diff --git a/src/input/schaum28.input.pamphlet b/src/input/schaum28.input.pamphlet index 5c68b06..3ba0bfc 100644 --- a/src/input/schaum28.input.pamphlet +++ b/src/input/schaum28.input.pamphlet @@ -234,6 +234,10 @@ aa:=integrate(x/cosh(a*x),x) \section{\cite{1}:14.569~~~~~$\displaystyle \int{\cosh^2{ax}}~dx$} $$\int{\cosh^2{ax}}= +\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2a} +$$ +Note that the Schaums print edition (1968 printing 3) has a typo: +$$\int{\cosh^2{ax}}= \frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2} $$ <<*>>= @@ -250,40 +254,21 @@ aa:=integrate(cosh(a*x)^2,x) --E --S 18 -bb:=x/2+(sinh(a*x)*cosh(a*x))/2 +bb:=x/2+(sinh(a*x)*cosh(a*x))/(2*a) --R ---R cosh(a x)sinh(a x) + x ---R (2) ---------------------- ---R 2 +--R cosh(a x)sinh(a x) + a x +--R (2) ------------------------ +--R 2a --R Type: Expression Integer --E ---S 19 +--S 19 14:569 Schaums and Axiom agree cc:=aa-bb --R ---R (- a + 1)cosh(a x)sinh(a x) ---R (3) --------------------------- ---R 2a +--R (3) 0 --R Type: Expression Integer --E ---S 20 -sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) ---R ---I %B sinh(y + x) - %B sinh(y - x) ---I (4) %B cosh(y)sinh(x) == ------------------------------- ---R 2 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 21 14:569 SCHAUMS AND AXIOM DISAGREE? -dd:=sinhcoshrule cc ---R ---R (- a + 1)sinh(2a x) ---R (5) ------------------- ---R 4a ---R Type: Expression Integer ---E @ \section{\cite{1}:14.570~~~~~$\displaystyle @@ -294,7 +279,7 @@ $$ <<*>>= )clear all ---S 22 +--S 20 aa:=integrate(x*cosh(a*x)^2,x) --R --R @@ -306,7 +291,7 @@ aa:=integrate(x*cosh(a*x)^2,x) --R Type: Union(Expression Integer,...) --E ---S 23 +--S 21 bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2) --R --R 2 2 @@ -317,7 +302,7 @@ bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2) --R Type: Expression Integer --E ---S 24 +--S 22 cc:=aa-bb --R --R (3) @@ -332,7 +317,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 25 +--S 23 sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R --R 2 cosh(2x) - 1 @@ -341,7 +326,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 26 +--S 24 dd:=sinhsqrrule cc --R --R (5) @@ -353,7 +338,7 @@ dd:=sinhsqrrule cc --R Type: Expression Integer --E ---S 27 +--S 25 coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R --R 2 cosh(2x) + 1 @@ -362,7 +347,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 28 +--S 26 ee:=coshsqrrule dd --R --R - x sinh(2a x) + 2x cosh(a x)sinh(a x) @@ -371,7 +356,7 @@ ee:=coshsqrrule dd --R Type: Expression Integer --E ---S 29 +--S 27 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %S sinh(y + x) - %S sinh(y - x) @@ -380,7 +365,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 30 14:570 Schaums and Axiom agree +--S 28 14:570 Schaums and Axiom agree ff:=sinhcoshrule ee --R --R (9) 0 @@ -396,7 +381,7 @@ $$ <<*>>= )clear all ---S 31 +--S 29 aa:=integrate(1/cosh(a*x)^2,x) --R --R @@ -407,7 +392,7 @@ aa:=integrate(1/cosh(a*x)^2,x) --R Type: Union(Expression Integer,...) --E ---S 32 +--S 30 bb:=tanh(a*x)/a --R --R tanh(a x) @@ -416,7 +401,7 @@ bb:=tanh(a*x)/a --R Type: Expression Integer --E ---S 33 +--S 31 cc:=aa-bb --R --R 2 2 @@ -425,155 +410,12 @@ cc:=aa-bb --R 2 2 --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a --R Type: Expression Integer ---E - ---S 34 -tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) ---R ---R sinh(x) ---R (4) tanh(x) == ------- ---R cosh(x) ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 35 -dd:=tanhrule cc ---R ---R (5) ---R 3 2 2 ---R - sinh(a x) - 2cosh(a x)sinh(a x) + (- cosh(a x) - 1)sinh(a x) ---R + ---R - 2cosh(a x) ---R / ---R 2 2 3 ---R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x) ---R Type: Expression Integer ---E - ---S 36 -sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) ---R ---R 2 cosh(2x) - 1 ---R (6) sinh(x) == ------------ ---R 2 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 37 -ee:=sinhsqrrule dd ---R ---R (7) ---R 3 2 ---R - 2sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - 2cosh(a x)cosh(2a x) ---R + ---R - 2cosh(a x) ---R / ---R 2 3 ---R 4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x) ---R + ---R a cosh(a x) ---R Type: Expression Integer ---E - ---S 38 -coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) ---R ---R 2 cosh(2x) + 1 ---R (8) cosh(x) == ------------ ---R 2 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 39 -ff:=coshsqrrule ee ---R ---R (9) ---R 3 ---R - 2sinh(a x) + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x) ---R + ---R - 2cosh(a x) ---R / ---R 3 ---R (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x) ---R + ---R a cosh(a x) ---R Type: Expression Integer ---E - ---S 40 -sinhcuberule:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x)) ---R ---R 3 sinh(3x) - 3sinh(x) ---R (10) sinh(x) == ------------------- ---R 4 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 41 -gg:=sinhcuberule ff ---R ---R (11) ---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x) ---R + ---R - 4cosh(a x) ---R / ---R 3 ---R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(a x)cosh(2a x) + 4a cosh(a x) ---R + ---R 2a cosh(a x) ---R Type: Expression Integer ---E - ---S 42 -coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) ---R ---R 3 cosh(3x) - 3cosh(x) ---R (12) cosh(x) == ------------------- ---R 4 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 43 -hh:=coshcuberule gg ---R ---R (13) ---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x) ---R + ---R - 4cosh(a x) ---R / ---R (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) + 2a cosh(a x)cosh(2a x) ---R + ---R - a cosh(a x) ---R Type: Expression Integer ---E - ---S 44 -coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) ---R ---I %U cosh(y + x) + %U cosh(y - x) ---I (14) %U cosh(x)cosh(y) == ------------------------------- ---R 2 ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 45 -ii:=coshcoshrule hh ---R ---R (15) ---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 2cosh(3a x) - 6cosh(a x) ---R ---------------------------------------------------------------------- ---R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(3a x) ---R Type: Expression Integer ---E - ---S 46 14:571 SCHAUMS AND AXIOM DISAGREE? -jj:=complexNormalize ii +--S 32 +dd:=complexNormalize cc --R ---R a x 2 ---R - 2(%e ) - 2 ---R (16) --------------- ---R a x 2 ---R 2a (%e ) - a +--R 1 +--R (4) - - +--R a --R Type: Expression Integer --E @ @@ -586,7 +428,7 @@ $$ <<*>>= )clear all ---S 47 +--S 33 aa:=integrate(cosh(a*x)*cosh(p*x),x) --R --R @@ -597,7 +439,7 @@ aa:=integrate(cosh(a*x)*cosh(p*x),x) --R Type: Union(Expression Integer,...) --E ---S 48 +--S 34 bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p)) --R --R (p - a)x sinh(p + a) + (p + a)x sinh(p - a) @@ -607,7 +449,7 @@ bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p)) --R Type: Expression Integer --E ---S 49 +--S 35 cc:=aa-bb --R --R (3) @@ -627,7 +469,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 50 +--S 36 sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R --R 2 cosh(2x) - 1 @@ -636,7 +478,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 51 +--S 37 dd:=sinhsqrrule cc --R --R (5) @@ -653,7 +495,7 @@ dd:=sinhsqrrule cc --R Type: Expression Integer --E ---S 52 +--S 38 coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R --R 2 cosh(2x) + 1 @@ -662,7 +504,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 53 +--S 39 ee:=coshsqrrule dd --R --R (7) @@ -675,7 +517,7 @@ ee:=coshsqrrule dd --R Type: Expression Integer --E ---S 54 +--S 40 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %V sinh(y + x) - %V sinh(y - x) @@ -684,7 +526,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 55 14:572 Schaums and Axiom agree +--S 41 14:572 Schaums and Axiom agree ff:=sinhcoshrule ee --R --R (9) @@ -706,7 +548,7 @@ $$ <<*>>= )clear all ---S 56 +--S 42 aa:=integrate(cosh(a*x)*sin(p*x),x) --R --R @@ -724,7 +566,7 @@ aa:=integrate(cosh(a*x)*sin(p*x),x) --R Type: Union(Expression Integer,...) --E ---S 57 +--S 43 bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2) --R --R a sin(p x)sinh(a x) - p cos(p x)cosh(a x) @@ -734,7 +576,7 @@ bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2) --R Type: Expression Integer --E ---S 58 +--S 44 cc:=aa-bb --R --R (3) @@ -749,7 +591,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 59 +--S 45 coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R --R 2 cosh(2x) + 1 @@ -758,7 +600,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 60 +--S 46 dd:=coshsqrrule cc --R --R (5) @@ -772,7 +614,7 @@ dd:=coshsqrrule cc --R Type: Expression Integer --E ---S 61 +--S 47 sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R --R 2 cosh(2x) - 1 @@ -781,7 +623,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 62 14:573 Schaums and Axiom agree +--S 48 14:573 Schaums and Axiom agree ee:=sinhsqrrule dd --R --R (7) 0 @@ -797,7 +639,7 @@ $$ <<*>>= )clear all ---S 63 +--S 49 aa:=integrate(cosh(a*x)*cos(p*x),x) --R --R @@ -815,7 +657,7 @@ aa:=integrate(cosh(a*x)*cos(p*x),x) --R Type: Union(Expression Integer,...) --E ---S 64 +--S 50 bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2) --R --R a cos(p x)sinh(a x) + p cosh(a x)sin(p x) @@ -825,7 +667,7 @@ bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2) --R Type: Expression Integer --E ---S 65 +--S 51 cc:=aa-bb --R --R (3) @@ -840,7 +682,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 66 +--S 52 coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R --R 2 cosh(2x) + 1 @@ -849,7 +691,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 67 +--S 53 dd:=coshsqrrule cc --R --R (5) @@ -863,7 +705,7 @@ dd:=coshsqrrule cc --R Type: Expression Integer --E ---S 68 +--S 54 sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R --R 2 cosh(2x) - 1 @@ -872,7 +714,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 69 14:574 Schaums and Axiom agree +--S 55 14:574 Schaums and Axiom agree ee:=sinhsqrrule dd --R --R (7) 0 @@ -888,7 +730,7 @@ $$ <<*>>= )clear all ---S 70 +--S 56 aa:=integrate(1/(cosh(a*x)+1),x) --R --R @@ -898,7 +740,7 @@ aa:=integrate(1/(cosh(a*x)+1),x) --R Type: Union(Expression Integer,...) --E ---S 71 +--S 57 bb:=1/a*tanh((a*x)/2) --R --R a x @@ -909,7 +751,7 @@ bb:=1/a*tanh((a*x)/2) --R Type: Expression Integer --E ---S 72 +--S 58 cc:=aa-bb --R --R a x @@ -920,7 +762,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 73 +--S 59 tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) --R --R sinh(x) @@ -929,7 +771,7 @@ tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 74 +--S 60 dd:=tanhrule cc --R --R a x a x a x @@ -942,7 +784,7 @@ dd:=tanhrule cc --R Type: Expression Integer --E ---S 75 +--S 61 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %B sinh(y + x) - %B sinh(y - x) @@ -951,7 +793,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 76 +--S 62 ee:=sinhcoshrule dd --R --R 3a x a x a x a x @@ -964,7 +806,7 @@ ee:=sinhcoshrule dd --R Type: Expression Integer --E ---S 77 +--S 63 sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R --I %B cosh(y + x) - %B cosh(y - x) @@ -973,7 +815,7 @@ sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 78 +--S 64 ff:=sinhsinhrule ee --R --R 3a x a x 3a x a x @@ -986,7 +828,7 @@ ff:=sinhsinhrule ee --R Type: Expression Integer --E ---S 79 +--S 65 coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R --I %BC cosh(y + x) + %BC cosh(y - x) @@ -995,7 +837,7 @@ coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 80 14:575 Schaums and Axiom differ by a constant +--S 66 14:575 Schaums and Axiom differ by a constant gg:=coshcoshrule ff --R --R 1 @@ -1013,7 +855,7 @@ $$ <<*>>= )clear all ---S 81 +--S 67 aa:=integrate(1/(cosh(a*x)-1),x) --R --R @@ -1023,7 +865,7 @@ aa:=integrate(1/(cosh(a*x)-1),x) --R Type: Union(Expression Integer,...) --E ---S 82 +--S 68 bb:=-1/a*coth((a*x)/2) --R --R a x @@ -1034,7 +876,7 @@ bb:=-1/a*coth((a*x)/2) --R Type: Expression Integer --E ---S 83 +--S 69 cc:=aa-bb --R --R a x a x @@ -1045,7 +887,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 84 +--S 70 cothrule:=rule(coth(x) == cosh(x)/sinh(x)) --R --R cosh(x) @@ -1054,7 +896,7 @@ cothrule:=rule(coth(x) == cosh(x)/sinh(x)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 85 +--S 71 dd:=cothrule cc --R --R a x a x a x a x @@ -1067,7 +909,7 @@ dd:=cothrule cc --R Type: Expression Integer --E ---S 86 +--S 72 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %BD sinh(y + x) - %BD sinh(y - x) @@ -1076,7 +918,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 87 +--S 73 ee:=sinhcoshrule dd --R --R 3a x a x a x a x @@ -1089,7 +931,7 @@ ee:=sinhcoshrule dd --R Type: Expression Integer --E ---S 88 +--S 74 sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R --I %BE cosh(y + x) - %BE cosh(y - x) @@ -1098,7 +940,7 @@ sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 89 +--S 75 ff:=sinhsinhrule ee --R --R 3a x a x a x a x @@ -1111,7 +953,7 @@ ff:=sinhsinhrule ee --R Type: Expression Integer --E ---S 90 +--S 76 coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R --I %BF cosh(y + x) + %BF cosh(y - x) @@ -1120,7 +962,7 @@ coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 91 14:576 Schaums and Axiom differ by a constant +--S 77 14:576 Schaums and Axiom differ by a constant gg:=coshcoshrule ff --R --R 1 @@ -1138,7 +980,7 @@ $$ <<*>>= )clear all ---S 92 +--S 78 aa:=integrate(x/(cosh(a*x)+1),x) --R --R @@ -1152,7 +994,7 @@ aa:=integrate(x/(cosh(a*x)+1),x) --R Type: Union(Expression Integer,...) --E ---S 93 +--S 79 bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2)) --R --R a x a x @@ -1164,7 +1006,7 @@ bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2)) --R Type: Expression Integer --E ---S 94 +--S 80 cc:=aa-bb --R --R (3) @@ -1185,7 +1027,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 95 +--S 81 tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) --R --R sinh(x) @@ -1194,7 +1036,7 @@ tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 96 +--S 82 dd:=tanhrule cc --R --R (5) @@ -1222,7 +1064,7 @@ dd:=tanhrule cc --R Type: Expression Integer --E ---S 97 +--S 83 coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R --I %BG cosh(y + x) + %BG cosh(y - x) @@ -1231,7 +1073,7 @@ coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 98 +--S 84 ee:=coshcoshrule dd --R --R (7) @@ -1259,7 +1101,7 @@ ee:=coshcoshrule dd --R Type: Expression Integer --E ---S 99 +--S 85 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %BH sinh(y + x) - %BH sinh(y - x) @@ -1268,7 +1110,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 100 +--S 86 ff:=sinhcoshrule ee --R --R (9) @@ -1296,7 +1138,7 @@ ff:=sinhcoshrule ee --R Type: Expression Integer --E ---S 101 +--S 87 sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R --I %BI cosh(y + x) - %BI cosh(y - x) @@ -1305,7 +1147,7 @@ sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 102 +--S 88 gg:=sinhsinhrule ff --R --R a x @@ -1317,7 +1159,7 @@ gg:=sinhsinhrule ff --R Type: Expression Integer --E ---S 103 14:577 Schaums and Axiom differ by a constant +--S 89 14:577 Schaums and Axiom differ by a constant complexNormalize gg --R --R 2log(2) @@ -1336,7 +1178,7 @@ $$ <<*>>= )clear all ---S 104 +--S 90 aa:=integrate(x/(cosh(a*x)-1),x) --R --R @@ -1350,7 +1192,7 @@ aa:=integrate(x/(cosh(a*x)-1),x) --R Type: Union(Expression Integer,...) --E ---S 105 +--S 91 bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2)) --R --R a x a x @@ -1362,7 +1204,7 @@ bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2)) --R Type: Expression Integer --E ---S 106 +--S 92 cc:=aa-bb --R --R (3) @@ -1383,7 +1225,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 107 +--S 93 cothrule:=rule(coth(x) == cosh(x)/sinh(x)) --R --R cosh(x) @@ -1392,7 +1234,7 @@ cothrule:=rule(coth(x) == cosh(x)/sinh(x)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 108 +--S 94 dd:=cothrule cc --R --R (5) @@ -1420,7 +1262,7 @@ dd:=cothrule cc --R Type: Expression Integer --E ---S 109 +--S 95 sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R --I %BJ sinh(y + x) - %BJ sinh(y - x) @@ -1429,7 +1271,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 110 +--S 96 ee:=sinhcoshrule dd --R --R (7) @@ -1457,7 +1299,7 @@ ee:=sinhcoshrule dd --R Type: Expression Integer --E ---S 111 +--S 97 sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R --I %BK cosh(y + x) - %BK cosh(y - x) @@ -1466,7 +1308,7 @@ sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 112 +--S 98 ff:=sinhsinhrule ee --R --R (9) @@ -1494,7 +1336,7 @@ ff:=sinhsinhrule ee --R Type: Expression Integer --E ---S 113 +--S 99 coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R --I %BL cosh(y + x) + %BL cosh(y - x) @@ -1503,7 +1345,7 @@ coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 114 +--S 100 gg:=coshcoshrule ff --R --R a x @@ -1515,7 +1357,7 @@ gg:=coshcoshrule ff --R Type: Expression Integer --E ---S 115 14:578 Schaums and Axiom differ by a constant +--S 101 14:578 Schaums and Axiom differ by a constant hh:=complexNormalize gg --R --R 2log(2) @@ -1534,7 +1376,7 @@ $$ <<*>>= )clear all ---S 116 +--S 102 aa:=integrate(1/(cosh(a*x)+1)^2,x) --R --R @@ -1552,7 +1394,7 @@ aa:=integrate(1/(cosh(a*x)+1)^2,x) --R Type: Union(Expression Integer,...) --E ---S 117 +--S 103 bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3 --R --R a x 3 a x @@ -1563,7 +1405,7 @@ bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3 --R Type: Expression Integer --E ---S 118 14:579 Axiom cannot compute this integral +--S 104 14:579 Axiom cannot compute this integral cc:=aa-bb --R --R (3) @@ -1614,7 +1456,7 @@ $$ <<*>>= )clear all ---S 119 +--S 105 aa:=integrate(1/(cosh(a*x)-1)^2,x) --R --R @@ -1632,7 +1474,7 @@ aa:=integrate(1/(cosh(a*x)-1)^2,x) --R Type: Union(Expression Integer,...) --E ---S 120 +--S 106 bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3 --R --R a x 3 a x @@ -1643,7 +1485,7 @@ bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3 --R Type: Expression Integer --E ---S 121 14:580 Axiom cannot compute this integral +--S 107 14:580 Axiom cannot compute this integral cc:=aa-bb --R --R (3) @@ -1702,7 +1544,7 @@ $$ <<*>>= )clear all ---S 122 +--S 108 aa:=integrate(1/(p+q*cosh(a*x)),x) --R --R @@ -1744,7 +1586,7 @@ aa:=integrate(1/(p+q*cosh(a*x)),x) --R Type: Union(List Expression Integer,...) --E ---S 123 +--S 109 bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2)) --R --R a x @@ -1760,7 +1602,7 @@ bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2)) --R Type: Expression Integer --E ---S 124 +--S 110 bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2-q^2))) --R --R +---------+ @@ -1777,7 +1619,7 @@ bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2 --R Type: Expression Integer --E ---S 125 +--S 111 cc1:=aa.1-bb1 --R --R (4) @@ -1817,7 +1659,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 126 +--S 112 cc2:=aa.2-bb1 --R --R +-------+ @@ -1834,7 +1676,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 127 +--S 113 cc3:=aa.1-bb2 --R --R (6) @@ -1871,7 +1713,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 128 14:581 Axiom cannot simplify this expression +--S 114 14:581 Axiom cannot simplify this expression cc4:=aa.2-bb2 --R --R (7) @@ -1907,7 +1749,7 @@ $$ <<*>>= )clear all ---S 129 +--S 115 aa:=integrate(1/(p+q*cosh(a*x))^2,x) --R --R @@ -1989,7 +1831,7 @@ aa:=integrate(1/(p+q*cosh(a*x))^2,x) --R Type: Union(List Expression Integer,...) --E ---S 130 +--S 116 t1:=integrate(1/(p+q*cosh(a*x)),x) --R --R (2) @@ -2030,7 +1872,7 @@ t1:=integrate(1/(p+q*cosh(a*x)),x) --R Type: Union(List Expression Integer,...) --E ---S 131 +--S 117 bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1 --R --R (3) @@ -2066,7 +1908,7 @@ bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1 --R Type: Expression Integer --E ---S 132 +--S 118 bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2 --R --R (4) @@ -2087,7 +1929,7 @@ bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2 --R Type: Expression Integer --E ---S 133 +--S 119 cc1:=aa.1-bb1 --R --R (5) @@ -2183,7 +2025,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 134 +--S 120 cc2:=aa.2-bb1 --R --R (6) @@ -2271,7 +2113,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 135 +--S 121 cc3:=aa.1-bb2 --R --R (7) @@ -2359,7 +2201,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 136 14:582 Axiom cannot simplify this expression +--S 122 14:582 Axiom cannot simplify this expression cc4:=aa.2-bb2 --R --R (8) @@ -2409,7 +2251,7 @@ $$ <<*>>= )clear all ---S 137 +--S 123 aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x) --R --R @@ -2477,7 +2319,7 @@ aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x) --R Type: Union(List Expression Integer,...) --E ---S 138 +--S 124 bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p^2-q^2))) --R --R +---------+ @@ -2494,7 +2336,7 @@ bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p --R Type: Expression Integer --E ---S 139 +--S 125 bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2)) --R --R p tanh(a x) @@ -2509,7 +2351,7 @@ bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2)) --R Type: Expression Integer --E ---S 140 +--S 126 cc1:=aa.1-bb1 --R --R (4) @@ -2562,7 +2404,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 141 +--S 127 cc2:=aa.2-bb1 --R --R (5) @@ -2599,7 +2441,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 142 +--S 128 cc3:=aa.1-bb2 --R --R (6) @@ -2658,7 +2500,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 143 14:583 Axiom cannot simplify this expression +--S 129 14:583 Axiom cannot simplify this expression cc4:=aa.2-bb2 --R --R (7) @@ -2708,7 +2550,7 @@ $$ <<*>>= )clear all ---S 144 +--S 130 aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x) --R --R @@ -2754,7 +2596,7 @@ aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x) --R Type: Union(Expression Integer,...) --E ---S 145 +--S 131 bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p^2+q^2))) --R --R +-------+ @@ -2771,7 +2613,7 @@ bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p --R Type: Expression Integer --E ---S 146 +--S 132 bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2)) --R --R p tanh(a x) @@ -2786,7 +2628,7 @@ bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2)) --R Type: Expression Integer --E ---S 147 +--S 133 cc1:=aa-bb1 --R --R (4) @@ -2839,7 +2681,7 @@ cc1:=aa-bb1 --R Type: Expression Integer --E ---S 148 14:584 Axiom cannot compute this integral +--S 134 14:584 Axiom cannot compute this integral cc2:=aa-bb2 --R --R (5) @@ -2899,7 +2741,7 @@ $$ <<*>>= )clear all ---S 149 14:585 Axiom cannot compute this integral +--S 135 14:585 Axiom cannot compute this integral aa:=integrate(x^m*cosh(a*x),x) --R --R @@ -2919,7 +2761,7 @@ $$ <<*>>= )clear all ---S 150 14:586 Axiom cannot compute this integral +--S 136 14:586 Axiom cannot compute this integral aa:=integrate(cosh(a*x)^n,x) --R --R @@ -2940,7 +2782,7 @@ $$ <<*>>= )clear all ---S 151 14:587 Axiom cannot compute this integral +--S 137 14:587 Axiom cannot compute this integral aa:=integrate(cosh(a*x)/x^n,x) --R --R @@ -2962,7 +2804,7 @@ $$ <<*>>= )clear all ---S 152 14:588 Axiom cannot compute this integral +--S 138 14:588 Axiom cannot compute this integral aa:=integrate(1/cosh(a*x)^n,x) --R --R @@ -2985,7 +2827,7 @@ $$ <<*>>= )clear all ---S 153 14:589 Axiom cannot compute this integral +--S 139 14:589 Axiom cannot compute this integral aa:=integrate(1/cosh(a*x)^n,x) --R --R diff --git a/src/input/schaum29.input.pamphlet b/src/input/schaum29.input.pamphlet index 37e2800..1aff49f 100644 --- a/src/input/schaum29.input.pamphlet +++ b/src/input/schaum29.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 14 +--S 1 aa:=integrate(sinh(a*x)*cosh(a*x),x) --R --R @@ -28,6 +28,63 @@ aa:=integrate(sinh(a*x)*cosh(a*x),x) --R 4a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=sinh(a*x)^2/(2*a) +--R +--R 2 +--R sinh(a x) +--R (2) ---------- +--R 2a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2 2 +--R - sinh(a x) + cosh(a x) +--R (3) ------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 4 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=sinhsqrrule cc +--R +--R 2 +--R - cosh(2a x) + 2cosh(a x) + 1 +--R (5) ------------------------------ +--R 8a +--R Type: Expression Integer +--E + +--S 6 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 7 14:590 Schaums and Axiom agree +ee:=coshsqrrule dd +--R +--R 1 +--R (7) -- +--R 4a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.591~~~~~$\displaystyle @@ -38,7 +95,7 @@ $$ <<*>>= )clear all ---S 2 of 14 +--S 8 aa:=integrate(sinh(p*x)*cosh(q*x),x) --R --R @@ -48,6 +105,35 @@ aa:=integrate(sinh(p*x)*cosh(q*x),x) --R (q - p )sinh(p x) + (- q + p )cosh(p x) --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=(cosh(p+q)*x)/(2*(p+q))+(cosh(p-q)*x)/(2*(p-q)) +--R +--R (q - p)x cosh(q + p) + (- q - p)x cosh(q - p) +--R (2) --------------------------------------------- +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E + +--S 10 14:591 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R - 2q sinh(p x)sinh(q x) +--R + +--R 2 +--R ((- q + p)x cosh(q + p) + (q + p)x cosh(q - p))sinh(p x) +--R + +--R 2p cosh(p x)cosh(q x) +--R + +--R 2 +--R ((q - p)x cosh(q + p) + (- q - p)x cosh(q - p))cosh(p x) +--R / +--R 2 2 2 2 2 2 +--R (2q - 2p )sinh(p x) + (- 2q + 2p )cosh(p x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.592~~~~~$\displaystyle @@ -58,7 +144,7 @@ $$ <<*>>= )clear all ---S 3 of 14 +--S 11 aa:=integrate(sinh(a*x)^n*cosh(a*x),x) --R --R @@ -68,6 +154,29 @@ aa:=integrate(sinh(a*x)^n*cosh(a*x),x) --R (a n + a)sinh(a x) + (- a n - a)cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 12 +bb:=sinh(a*x)/((n+1)*a) +--R +--R sinh(a x) +--R (2) --------- +--R a n + a +--R Type: Expression Integer +--E + +--S 13 14:592 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R - sinh(a x)sinh(n log(sinh(a x))) - sinh(a x)cosh(n log(sinh(a x))) +--R + +--R 3 2 +--R - sinh(a x) + cosh(a x) sinh(a x) +--R / +--R 2 2 +--R (a n + a)sinh(a x) + (- a n - a)cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.593~~~~~$\displaystyle @@ -78,7 +187,7 @@ $$ <<*>>= )clear all ---S 4 of 14 +--S 14 aa:=integrate(cosh(a*x)^n*sinh(a*x),x) --R --R @@ -88,6 +197,29 @@ aa:=integrate(cosh(a*x)^n*sinh(a*x),x) --R (a n + a)sinh(a x) + (- a n - a)cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 15 +bb:=cosh(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R cosh(a x) +--R (2) -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 16 14:593 Axiom cannot simplify this expression +--R +--R (3) +--R - cosh(a x)sinh(n log(cosh(a x))) - cosh(a x)cosh(n log(cosh(a x))) +--R + +--R 2 2 n + 1 +--R (- sinh(a x) + cosh(a x) )cosh(a x) +--R / +--R 2 2 +--R (a n + a)sinh(a x) + (- a n - a)cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.594~~~~~$\displaystyle @@ -98,7 +230,7 @@ $$ <<*>>= )clear all ---S 5 of 14 +--S 17 aa:=integrate(sinh(a*x)^2*cosh(a*x)^2,x) --R --R @@ -108,6 +240,22 @@ aa:=integrate(sinh(a*x)^2*cosh(a*x)^2,x) --R 8a --R Type: Union(Expression Integer,...) --E + +--S 18 +bb:=sinh(4*a*x)/(32*a)-x/8 +--R +--R sinh(4a x) - 4a x +--R (2) ----------------- +--R 32a +--R Type: Expression Integer +--E + +--S 19 14:594 Schaums and Axiom agree +cc:=complexNormalize(aa-bb) +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.595~~~~~$\displaystyle @@ -118,7 +266,7 @@ $$ <<*>>= )clear all ---S 6 of 14 +--S 20 aa:=integrate(1/(sinh(a*x)*cosh(a*x)),x) --R --R @@ -129,6 +277,67 @@ aa:=integrate(1/(sinh(a*x)*cosh(a*x)),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 21 +bb:=1/a*log(tanh(a*x)) +--R +--R log(tanh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R (3) +--R 2cosh(a x) +--R - log(tanh(a x)) - log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 23 +dd:=expandLog cc +--R +--R - log(tanh(a x)) + log(sinh(a x)) - log(cosh(a x)) +--R (4) -------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 24 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (5) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +ee:=tanhrule dd +--R +--R sinh(a x) +--R log(sinh(a x)) - log(---------) - log(cosh(a x)) +--R cosh(a x) +--R (6) ------------------------------------------------ +--R a +--R Type: Expression Integer +--E + +--S 26 14:595 Schaums and Axiom agree +ff:=expandLog ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.596~~~~~$\displaystyle @@ -139,17 +348,52 @@ $$ <<*>>= )clear all ---S 7 of 14 -aa:=integrate(1/(sinh(a*x)^2*cos(a*x)),x) ---R +--S 27 +aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)),x) --R ---R x ---R ++ 1 ---I (1) | -------------------- d%R ---R ++ 2 ---I cos(%R a)sinh(%R a) +--R (1) +--R 2 2 +--R (- 2sinh(a x) - 4cosh(a x)sinh(a x) - 2cosh(a x) + 2) +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R - 2sinh(a x) - 2cosh(a x) +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 28 +bb:=-1/a*atan(sinh(a*x)-csch(a*x))/a +--R +--R atan(sinh(a x) - csch(a x)) +--R (2) - --------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 29 14:596 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- 2a sinh(a x) - 4a cosh(a x)sinh(a x) - 2a cosh(a x) + 2a) +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R atan(sinh(a x) - csch(a x)) +--R + +--R - 2a sinh(a x) - 2a cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.597~~~~~$\displaystyle @@ -160,7 +404,7 @@ $$ <<*>>= )clear all ---S 8 of 14 +--S 30 aa:=integrate(1/(sinh(a*x)*cosh(a*x)^2),x) --R --R @@ -181,6 +425,272 @@ aa:=integrate(1/(sinh(a*x)*cosh(a*x)^2),x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 31 +bb:=sech(a*x)/a+1/a*log(tanh((a*x)/2)) +--R +--R a x +--R log(tanh(---)) + sech(a x) +--R 2 +--R (2) -------------------------- +--R a +--R Type: Expression Integer +--E + +--S 32 +cc:=aa-bb +--R +--R (3) +--R 2 2 a x +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)log(tanh(---)) +--R 2 +--R + +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R - sech(a x)sinh(a x) + (- 2cosh(a x)sech(a x) + 2)sinh(a x) +--R + +--R 2 +--R (- cosh(a x) - 1)sech(a x) + 2cosh(a x) +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 33 +sechrule:=rule(sech(x) == 1/cosh(x)) +--R +--R 1 +--R (4) sech(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 +dd:=sechrule cc +--R +--R (5) +--R 2 2 3 +--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x)) +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 2 3 +--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 3 +--R (cosh(a x)sinh(a x) + 2cosh(a x) sinh(a x) + cosh(a x) + cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - sinh(a x) + cosh(a x) - 1 +--R / +--R 2 2 3 +--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x) +--R Type: Expression Integer +--E + +--S 35 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (6) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +ee:=tanhrule dd +--R +--R (7) +--R 2 2 3 +--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 3 +--R (cosh(a x)sinh(a x) + 2cosh(a x) sinh(a x) + cosh(a x) + cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 3 +--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x)) +--R * +--R a x +--R sinh(---) +--R 2 +--R log(---------) +--R a x +--R cosh(---) +--R 2 +--R + +--R 2 2 +--R - sinh(a x) + cosh(a x) - 1 +--R / +--R 2 2 3 +--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x) +--R Type: Expression Integer +--E + +--S 37 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (8) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +ff:=coshcuberule ee +--R +--R (9) +--R 2 2 +--R - 4cosh(a x)sinh(a x) - 8cosh(a x) sinh(a x) - cosh(3a x) +--R + +--R - cosh(a x) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (4cosh(a x)sinh(a x) + 8cosh(a x) sinh(a x) + cosh(3a x) + cosh(a x)) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - 4cosh(a x)sinh(a x) - 8cosh(a x) sinh(a x) - cosh(3a x) +--R + +--R - cosh(a x) +--R * +--R a x +--R sinh(---) +--R 2 +--R log(---------) +--R a x +--R cosh(---) +--R 2 +--R + +--R 2 2 +--R - 4sinh(a x) + 4cosh(a x) - 4 +--R / +--R 2 2 +--R 4a cosh(a x)sinh(a x) + 8a cosh(a x) sinh(a x) + a cosh(3a x) +--R + +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 39 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (10) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 40 +gg:=coshsqrrule ff +--R +--R (11) +--R 2 +--R - 4cosh(a x)sinh(a x) + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a x) +--R + +--R - cosh(a x) +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R 4cosh(a x)sinh(a x) + (4cosh(2a x) + 4)sinh(a x) + cosh(3a x) +--R + +--R cosh(a x) +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R - 4cosh(a x)sinh(a x) + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a x) +--R + +--R - cosh(a x) +--R * +--R a x +--R sinh(---) +--R 2 +--R log(---------) +--R a x +--R cosh(---) +--R 2 +--R + +--R 2 +--R - 4sinh(a x) + 2cosh(2a x) - 2 +--R / +--R 2 +--R 4a cosh(a x)sinh(a x) + (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) +--R + +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 41 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (12) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 42 +hh:=sinhsqrrule gg +--R +--R (13) +--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1) +--R + +--R a x +--R sinh(---) +--R 2 +--R - log(---------) +--R a x +--R cosh(---) +--R 2 +--R / +--R a +--R Type: Expression Integer +--E + +--S 43 +ii:=expandLog hh +--R +--R (14) +--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1) +--R + +--R a x a x +--R - log(sinh(---)) + log(cosh(---)) +--R 2 2 +--R / +--R a +--R Type: Expression Integer +--E + +--S 44 14:597 Schaums and Axiom agree +jj:=complexNormalize ii +--R +--R (15) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.598~~~~~$\displaystyle @@ -191,7 +701,7 @@ $$ <<*>>= )clear all ---S 9 of 14 +--S 45 aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)^2),x) --R --R @@ -206,6 +716,35 @@ aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)^2),x) --R 4a cosh(a x) sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 46 +bb:=-(2*coth(2*a*x))/a +--R +--R 2coth(2a x) +--R (2) - ----------- +--R a +--R Type: Expression Integer +--E + +--S 47 14:598 Axiom cannot simplify this expression +--R +--R (3) +--R 4 3 +--R 2coth(2a x)sinh(a x) + 8cosh(a x)coth(2a x)sinh(a x) +--R + +--R 2 2 3 +--R 12cosh(a x) coth(2a x)sinh(a x) + 8cosh(a x) coth(2a x)sinh(a x) +--R + +--R 4 +--R (2cosh(a x) - 2)coth(2a x) - 4 +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + 6a cosh(a x) sinh(a x) +--R + +--R 3 4 +--R 4a cosh(a x) sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.599~~~~~$\displaystyle @@ -216,7 +755,7 @@ $$ <<*>>= )clear all ---S 10 of 14 +--S 48 aa:=integrate(sinh(a*x)^2/cosh(a*x),x) --R --R @@ -230,6 +769,28 @@ aa:=integrate(sinh(a*x)^2/cosh(a*x),x) --R 2a sinh(a x) + 2a cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 49 +bb:=sinh(a*x)/a-1/a*atan(sinh(a*x)) +--R +--R - atan(sinh(a x)) + sinh(a x) +--R (2) ----------------------------- +--R a +--R Type: Expression Integer +--E + +--S 50 14:599 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R (- 4sinh(a x) - 4cosh(a x))atan(sinh(a x) + cosh(a x)) +--R + +--R 2 2 +--R (2sinh(a x) + 2cosh(a x))atan(sinh(a x)) - sinh(a x) + cosh(a x) - 1 +--R / +--R 2a sinh(a x) + 2a cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.600~~~~~$\displaystyle @@ -240,7 +801,7 @@ $$ <<*>>= )clear all ---S 11 of 14 +--S 51 aa:=integrate(cosh(a*x)^2/sinh(a*x),x) --R --R @@ -256,6 +817,37 @@ aa:=integrate(cosh(a*x)^2/sinh(a*x),x) --R 2a sinh(a x) + 2a cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=cosh(a*x)/a+1/a*log(tanh((a*x)/2)) +--R +--R a x +--R log(tanh(---)) + cosh(a x) +--R 2 +--R (2) -------------------------- +--R a +--R Type: Expression Integer +--E + +--S 53 14:600 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R a x +--R (- 2sinh(a x) - 2cosh(a x))log(tanh(---)) +--R 2 +--R + +--R (- 2sinh(a x) - 2cosh(a x))log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (2sinh(a x) + 2cosh(a x))log(sinh(a x) + cosh(a x) - 1) + sinh(a x) +--R + +--R 2 +--R - cosh(a x) + 1 +--R / +--R 2a sinh(a x) + 2a cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.601~~~~~$\displaystyle @@ -267,7 +859,7 @@ $$ <<*>>= )clear all ---S 12 of 14 +--S 54 aa:=integrate(1/(cosh(a*x)*(1+sinh(a*x))),x) --R --R @@ -281,6 +873,87 @@ aa:=integrate(1/(cosh(a*x)*(1+sinh(a*x))),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 55 +bb:=1/(2*a)*log((1+sinh(a*x))/cosh(a*x))+1/a*atan(%e^(a*x)) +--R +--R sinh(a x) + 1 a x +--R log(-------------) + 2atan(%e ) +--R cosh(a x) +--R (2) --------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R (3) +--R sinh(a x) + 1 2cosh(a x) +--R - log(-------------) - log(- ---------------------) +--R cosh(a x) sinh(a x) - cosh(a x) +--R + +--R - 2sinh(a x) - 2 a x +--R log(---------------------) + 2atan(sinh(a x) + cosh(a x)) - 2atan(%e ) +--R sinh(a x) - cosh(a x) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 57 +dd:=expandLog cc +--R +--R a x +--R atan(sinh(a x) + cosh(a x)) - atan(%e ) +--R (4) ----------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 58 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (5) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 59 +ee:=atanrule dd +--R +--R a x +--R - %e + %i - sinh(a x) - cosh(a x) + %i +--R %i log(------------) - %i log(----------------------------) +--R a x sinh(a x) + cosh(a x) + %i +--R %e + %i +--R (6) ----------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 60 +ff:=expandLog ee +--R +--R (7) +--R %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) - %i) +--R + +--R a x a x +--R - %i log(%e + %i) + %i log(%e - %i) +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 61 14:601 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (8) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.602~~~~~$\displaystyle @@ -291,7 +964,7 @@ $$ <<*>>= )clear all ---S 13 of 14 +--S 62 aa:=integrate(1/(sinh(a*x)*(cosh(a*x)+1)),x) --R --R @@ -316,6 +989,204 @@ aa:=integrate(1/(sinh(a*x)*(cosh(a*x)+1)),x) --R 4a cosh(a x) + 2a --R Type: Union(Expression Integer,...) --E + +--S 63 +bb:=1/(2*a)*log(tanh((a*x)/2))+1/(2*a*(cosh(a*x)+1)) +--R +--R a x +--R (cosh(a x) + 1)log(tanh(---)) + 1 +--R 2 +--R (2) --------------------------------- +--R 2a cosh(a x) + 2a +--R Type: Expression Integer +--E + +--S 64 +cc:=aa-bb +--R +--R (3) +--R 2 +--R (- cosh(a x) - 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a x) +--R + +--R - 3cosh(a x) - 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- cosh(a x) - 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a x) +--R + +--R - 3cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (cosh(a x) + 1)sinh(a x) + (2cosh(a x) + 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) + 3cosh(a x) + 3cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - sinh(a x) + cosh(a x) - 1 +--R / +--R 2 +--R (2a cosh(a x) + 2a)sinh(a x) +--R + +--R 2 3 +--R (4a cosh(a x) + 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R 6a cosh(a x) + 6a cosh(a x) + 2a +--R Type: Expression Integer +--E + +--S 65 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (4) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 66 +dd:=coshcuberule cc +--R +--R (5) +--R 2 +--R (- 4cosh(a x) - 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R - 12cosh(a x) - 9cosh(a x) - 4 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- 4cosh(a x) - 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R - 12cosh(a x) - 9cosh(a x) - 4 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 2 +--R (4cosh(a x) + 4)sinh(a x) + (8cosh(a x) + 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) + 12cosh(a x) + 9cosh(a x) + 4 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R - 4sinh(a x) + 4cosh(a x) - 4 +--R / +--R 2 +--R (8a cosh(a x) + 8a)sinh(a x) +--R + +--R 2 +--R (16a cosh(a x) + 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R 24a cosh(a x) + 18a cosh(a x) + 8a +--R Type: Expression Integer +--E + +--S 67 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 68 +ee:=sinhsqrrule dd +--R +--R (7) +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) - 7cosh(a x) - 2 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) - 7cosh(a x) - 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (8cosh(a x) + 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) + 2)cosh(2a x) + 12cosh(a x) + 7cosh(a x) + 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R - 2cosh(2a x) + 4cosh(a x) - 2 +--R / +--R 2 +--R (16a cosh(a x) + 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R (4a cosh(a x) + 4a)cosh(2a x) + 24a cosh(a x) + 14a cosh(a x) + 4a +--R Type: Expression Integer +--E + +--S 69 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (8) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 70 +ff:=coshsqrrule ee +--R +--R (9) +--R a x +--R - log(tanh(---)) - log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R log(sinh(a x) + cosh(a x) - 1) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 71 14:602 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (10) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.603~~~~~$\displaystyle @@ -326,7 +1197,7 @@ $$ <<*>>= )clear all ---S 14 of 14 +--S 72 aa:=integrate(1/(sinh(a*x)*(cosh(a*x)-1)),x) --R --R @@ -352,6 +1223,199 @@ aa:=integrate(1/(sinh(a*x)*(cosh(a*x)-1)),x) --R Type: Union(Expression Integer,...) --E +--S 73 +bb:=-1/(2*a)*log(tanh((a*x)/2))-1/(2*a*(cosh(a*x)-1)) +--R +--R a x +--R (- cosh(a x) + 1)log(tanh(---)) - 1 +--R 2 +--R (2) ----------------------------------- +--R 2a cosh(a x) - 2a +--R Type: Expression Integer +--E + +--S 74 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (cosh(a x) - 1)sinh(a x) + (2cosh(a x) - 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 2 +--R (cosh(a x) - 1)sinh(a x) + (2cosh(a x) - 4cosh(a x) + 2)sinh(a x) +--R + +--R 3 2 +--R cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- cosh(a x) + 1)sinh(a x) +--R + +--R 2 3 2 +--R (- 2cosh(a x) + 4cosh(a x) - 2)sinh(a x) - cosh(a x) + 3cosh(a x) +--R + +--R - 3cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R sinh(a x) - cosh(a x) + 1 +--R / +--R 2 +--R (2a cosh(a x) - 2a)sinh(a x) +--R + +--R 2 3 +--R (4a cosh(a x) - 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R - 6a cosh(a x) + 6a cosh(a x) - 2a +--R Type: Expression Integer +--E + +--S 75 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (4) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 76 +dd:=coshcuberule cc +--R +--R (5) +--R 2 2 +--R (4cosh(a x) - 4)sinh(a x) + (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) - 12cosh(a x) + 9cosh(a x) - 4 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 2 +--R (4cosh(a x) - 4)sinh(a x) + (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) +--R + +--R 2 +--R cosh(3a x) - 12cosh(a x) + 9cosh(a x) - 4 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- 4cosh(a x) + 4)sinh(a x) +--R + +--R 2 +--R (- 8cosh(a x) + 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R 12cosh(a x) - 9cosh(a x) + 4 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 2 +--R 4sinh(a x) - 4cosh(a x) + 4 +--R / +--R 2 +--R (8a cosh(a x) - 8a)sinh(a x) +--R + +--R 2 +--R (16a cosh(a x) - 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R - 24a cosh(a x) + 18a cosh(a x) - 8a +--R Type: Expression Integer +--E + +--S 77 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 78 +ee:=sinhsqrrule dd +--R +--R (7) +--R 2 +--R (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) + 7cosh(a x) - 2 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 2 +--R (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) + cosh(3a x) +--R + +--R 2 +--R (2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) + 7cosh(a x) - 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 2 +--R (- 8cosh(a x) + 16cosh(a x) - 8)sinh(a x) - cosh(3a x) +--R + +--R 2 +--R (- 2cosh(a x) + 2)cosh(2a x) + 12cosh(a x) - 7cosh(a x) + 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 2 +--R 2cosh(2a x) - 4cosh(a x) + 2 +--R / +--R 2 +--R (16a cosh(a x) - 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x) +--R + +--R 2 +--R (4a cosh(a x) - 4a)cosh(2a x) - 24a cosh(a x) + 14a cosh(a x) - 4a +--R Type: Expression Integer +--E + +--S 79 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (8) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 80 +ff:=coshsqrrule ee +--R +--R (9) +--R a x +--R log(tanh(---)) + log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R - log(sinh(a x) + cosh(a x) - 1) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 81 14:603 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (10) 0 +--R Type: Expression Integer +--E + )spool )lisp (bye) @ diff --git a/src/input/schaum30.input.pamphlet b/src/input/schaum30.input.pamphlet index 0e1aaf4..5df48bc 100644 --- a/src/input/schaum30.input.pamphlet +++ b/src/input/schaum30.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 11 +--S 1 aa:=integrate(tanh(a*x),x) --R --R @@ -29,6 +29,44 @@ aa:=integrate(tanh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/a*log(cosh(a*x)) +--R +--R log(cosh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2cosh(a x) +--R - log(cosh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 5 14:604 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.605~~~~~$\displaystyle @@ -39,7 +77,7 @@ $$ <<*>>= )clear all ---S 2 of 11 +--S 6 aa:=integrate(tanh(a*x)^2,x) --R --R @@ -48,6 +86,42 @@ aa:=integrate(tanh(a*x)^2,x) --R a cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 7 +bb:=x-tanh(a*x)/a +--R +--R - tanh(a x) + a x +--R (2) ----------------- +--R a +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R cosh(a x)tanh(a x) - sinh(a x) + cosh(a x) +--R (3) ------------------------------------------ +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 9 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 10 14:605 Schaums and Axiom differ by a constant +dd:=tanhrule cc +--R +--R 1 +--R (5) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.606~~~~~$\displaystyle @@ -58,7 +132,7 @@ $$ <<*>>= )clear all ---S 3 of 11 +--S 11 aa:=integrate(tanh(a*x)^3,x) --R --R @@ -92,6 +166,69 @@ aa:=integrate(tanh(a*x)^3,x) --R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 12 +bb:=1/a*log(cosh(a*x))-tanh(a*x)^2/(2*a) +--R +--R 2 +--R 2log(cosh(a x)) - tanh(a x) +--R (2) ---------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 13 14:606 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R - 2sinh(a x) - 8cosh(a x)sinh(a x) + (- 12cosh(a x) - 4)sinh(a x) +--R + +--R 3 4 2 +--R (- 8cosh(a x) - 8cosh(a x))sinh(a x) - 2cosh(a x) - 4cosh(a x) - 2 +--R * +--R log(cosh(a x)) +--R + +--R 4 3 2 2 +--R 2sinh(a x) + 8cosh(a x)sinh(a x) + (12cosh(a x) + 4)sinh(a x) +--R + +--R 3 4 2 +--R (8cosh(a x) + 8cosh(a x))sinh(a x) + 2cosh(a x) + 4cosh(a x) + 2 +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) + 4cosh(a x))sinh(a x) + cosh(a x) + 2cosh(a x) + 1 +--R * +--R 2 +--R tanh(a x) +--R + +--R 4 3 +--R - 2a x sinh(a x) - 8a x cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (- 12a x cosh(a x) - 4a x + 4)sinh(a x) +--R + +--R 3 4 +--R (- 8a x cosh(a x) + (- 8a x + 8)cosh(a x))sinh(a x) - 2a x cosh(a x) +--R + +--R 2 +--R (- 4a x + 4)cosh(a x) - 2a x +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) + 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) + 8a cosh(a x))sinh(a x) + 2a cosh(a x) + 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.607~~~~~$\displaystyle @@ -102,7 +239,7 @@ $$ <<*>>= )clear all ---S 4 of 11 +--S 14 aa:=integrate(tanh(a*x)^n*sech(a*x)^2,x) --R --R @@ -113,6 +250,31 @@ aa:=integrate(tanh(a*x)^n*sech(a*x)^2,x) --R (a n + a)cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 15 +bb:=tanh(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R tanh(a x) +--R (2) -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 16 14:607 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R sinh(a x) sinh(a x) +--R sinh(a x)sinh(n log(---------)) + sinh(a x)cosh(n log(---------)) +--R cosh(a x) cosh(a x) +--R + +--R n + 1 +--R - cosh(a x)tanh(a x) +--R / +--R (a n + a)cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.608~~~~~$\displaystyle @@ -123,7 +285,7 @@ $$ <<*>>= )clear all ---S 5 of 11 +--S 17 aa:=integrate(sech(a*x)^2/tanh(a*x),x) --R --R @@ -134,6 +296,63 @@ aa:=integrate(sech(a*x)^2/tanh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 18 +bb:=1/a*log(tanh(a*x)) +--R +--R log(tanh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 19 +cc:=aa-bb +--R +--R (3) +--R 2cosh(a x) +--R - log(tanh(a x)) - log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 20 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 21 +dd:=tanhrule cc +--R +--R (5) +--R sinh(a x) 2cosh(a x) +--R - log(---------) - log(- ---------------------) +--R cosh(a x) sinh(a x) - cosh(a x) +--R + +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 22 14:608 Schaums and Axiom agree +ee:=expandLog dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.609~~~~~$\displaystyle @@ -144,7 +363,7 @@ $$ <<*>>= )clear all ---S 6 of 11 +--S 23 aa:=integrate(1/tanh(a*x),x) --R --R @@ -155,6 +374,44 @@ aa:=integrate(1/tanh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=1/a*log(sinh(a*x)) +--R +--R log(sinh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2sinh(a x) +--R - log(sinh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 26 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 27 14:609 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.610~~~~~$\displaystyle @@ -167,7 +424,7 @@ $$ <<*>>= )clear all ---S 7 of 11 +--S 28 14:610 Axiom cannot compute this integral aa:=integrate(x*tanh(a*x),x) --R --R @@ -187,7 +444,7 @@ $$ <<*>>= )clear all ---S 8 of 11 +--S 29 aa:=integrate(x*tanh(a*x)^2,x) --R --R @@ -209,6 +466,159 @@ aa:=integrate(x*tanh(a*x)^2,x) --R 2a sinh(a x) + 4a cosh(a x)sinh(a x) + 2a cosh(a x) + 2a --R Type: Union(Expression Integer,...) --E + +--S 30 +bb:=x^2/2-(x*tanh(a*x))/a+1/a^2*log(cosh(a*x)) +--R +--R 2 2 +--R 2log(cosh(a x)) - 2a x tanh(a x) + a x +--R (2) --------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)log(cosh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R (a x sinh(a x) + 2a x cosh(a x)sinh(a x) + a x cosh(a x) + a x) +--R * +--R tanh(a x) +--R + +--R 2 2 +--R - 2a x sinh(a x) - 4a x cosh(a x)sinh(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 32 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 33 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) - 1)log(cosh(a x)) +--R + +--R 2 +--R (4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) + 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 +--R (4a x cosh(a x)sinh(a x) + a x cosh(2a x) + 2a x cosh(a x) + a x) +--R * +--R tanh(a x) +--R + +--R 2 +--R - 8a x cosh(a x)sinh(a x) - 2a x cosh(2a x) - 4a x cosh(a x) + 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 34 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 +ee:=coshsqrrule dd +--R +--R (7) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)log(cosh(a x)) +--R + +--R 2cosh(a x) +--R (2cosh(a x)sinh(a x) + cosh(2a x) + 1)log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R (2a x cosh(a x)sinh(a x) + a x cosh(2a x) + a x)tanh(a x) +--R + +--R - 4a x cosh(a x)sinh(a x) - 2a x cosh(2a x) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 36 +ff:=expandLog ee +--R +--R (8) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)log(sinh(a x) - cosh(a x)) +--R + +--R (2a x cosh(a x)sinh(a x) + a x cosh(2a x) + a x)tanh(a x) +--R + +--R (2log(- 2) - 4a x)cosh(a x)sinh(a x) + (log(- 2) - 2a x)cosh(2a x) +--R + +--R log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 37 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %N sinh(y + x) - %N sinh(y - x) +--I (9) %N cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +gg:=sinhcoshrule ff +--R +--R (10) +--R (- sinh(2a x) - cosh(2a x) - 1)log(sinh(a x) - cosh(a x)) +--R + +--R (a x sinh(2a x) + a x cosh(2a x) + a x)tanh(a x) +--R + +--R (log(- 2) - 2a x)sinh(2a x) + (log(- 2) - 2a x)cosh(2a x) + log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 39 14:611 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R - log(- 1) + log(- 2) +--R (11) --------------------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.612~~~~~$\displaystyle @@ -220,7 +630,7 @@ $$ <<*>>= )clear all ---S 9 of 11 +--S 40 14:612 Axiom cannot compute this integral aa:=integrate(tanh(a*x)/x,x) --R --R @@ -240,7 +650,7 @@ $$ <<*>>= )clear all ---S 10 of 11 +--S 41 aa:=integrate(1/(p+q*tanh(a*x)),x) --R --R @@ -252,6 +662,54 @@ aa:=integrate(1/(p+q*tanh(a*x)),x) --R a q - a p --R Type: Union(Expression Integer,...) --E + +--S 42 +bb:=(p*x)/(p^2-q^2)-q/(a*(p^2-q^2))*log(q*sinh(a*x)+p*cosh(a*x)) +--R +--R q log(q sinh(a x) + p cosh(a x)) - a p x +--R (2) ---------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (3) +--R - 2q sinh(a x) - 2p cosh(a x) +--R - q log(q sinh(a x) + p cosh(a x)) + q log(-----------------------------) +--R sinh(a x) - cosh(a x) +--R + +--R - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 44 +dd:=expandLog cc +--R +--R (4) +--R - q log(q sinh(a x) + p cosh(a x)) - q log(sinh(a x) - cosh(a x)) +--R + +--R q log(- q sinh(a x) - p cosh(a x)) + q log(2) - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 45 14:613 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R q log(2) - 2q log(- 1) +--R (5) ---------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.614~~~~~$\displaystyle @@ -262,7 +720,7 @@ $$ <<*>>= )clear all ---S 11 of 11 +--S 46 14:614 Axiom cannot compute this integral aa:=integrate(tanh(a*x)^n,x) --R --R diff --git a/src/input/schaum31.input.pamphlet b/src/input/schaum31.input.pamphlet index 3e81bf2..343aa2a 100644 --- a/src/input/schaum31.input.pamphlet +++ b/src/input/schaum31.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 11 +--S 1 aa:=integrate(coth(a*x),x) --R --R @@ -29,6 +29,44 @@ aa:=integrate(coth(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/a*log(sinh(a*x)) +--R +--R log(sinh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2sinh(a x) +--R - log(sinh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 5 14:615 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.616~~~~~$\displaystyle @@ -39,7 +77,7 @@ $$ <<*>>= )clear all ---S 2 of 11 +--S 6 aa:=integrate(coth(a*x)^2,x) --R --R @@ -48,6 +86,33 @@ aa:=integrate(coth(a*x)^2,x) --R a sinh(a x) --R Type: Union(Expression Integer,...) --E + +--S 7 +bb:=x-coth(a*x)/a +--R +--R - coth(a x) + a x +--R (2) ----------------- +--R a +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R (coth(a x) + 1)sinh(a x) - cosh(a x) +--R (3) ------------------------------------ +--R a sinh(a x) +--R Type: Expression Integer +--E + +--S 9 14:616 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 1 +--R (4) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.617~~~~~$\displaystyle @@ -58,7 +123,7 @@ $$ <<*>>= )clear all ---S 3 of 11 +--S 10 aa:=integrate(coth(a*x)^3,x) --R --R @@ -92,6 +157,69 @@ aa:=integrate(coth(a*x)^3,x) --R (4a cosh(a x) - 4a cosh(a x))sinh(a x) + a cosh(a x) - 2a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 11 +bb:=1/a*log(sinh(a*x)-coth(a*x)^2)/(2*a) +--R +--R 2 +--R log(sinh(a x) - coth(a x) ) +--R (2) --------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 12 14:617 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x) - 1 +--R * +--R 2 +--R log(sinh(a x) - coth(a x) ) +--R + +--R 4 3 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (12a cosh(a x) - 4a)sinh(a x) +--R + +--R 3 4 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) +--R + +--R 2 +--R - 4a cosh(a x) + 2a +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 4 2 3 +--R - 2a x sinh(a x) - 8a x cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 +--R (- 12a x cosh(a x) + 4a x - 4a)sinh(a x) +--R + +--R 2 3 2 2 4 +--R (- 8a x cosh(a x) + (8a x - 8a)cosh(a x))sinh(a x) - 2a x cosh(a x) +--R + +--R 2 2 2 +--R (4a x - 4a)cosh(a x) - 2a x +--R / +--R 2 4 2 3 2 2 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a )sinh(a x) +--R + +--R 2 3 2 2 4 2 2 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a x) +--R + +--R 2 +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.618~~~~~$\displaystyle @@ -102,7 +230,7 @@ $$ <<*>>= )clear all ---S 4 of 11 +--S 13 aa:=integrate(coth(a*x)^n*csch(a*x)^2,x) --R --R @@ -113,6 +241,53 @@ aa:=integrate(coth(a*x)^n*csch(a*x)^2,x) --R (a n + a)sinh(a x) --R Type: Union(Expression Integer,...) --E + +--S 14 +bb:=-coth(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R coth(a x) +--R (2) - -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R (3) +--R cosh(a x) cosh(a x) +--R - cosh(a x)sinh(n log(---------)) - cosh(a x)cosh(n log(---------)) +--R sinh(a x) sinh(a x) +--R + +--R n + 1 +--R sinh(a x)coth(a x) +--R / +--R (a n + a)sinh(a x) +--R Type: Expression Integer +--E + +--S 16 +dd:=expandLog cc +--R +--R (4) +--R cosh(a x)sinh(n log(sinh(a x)) - n log(cosh(a x))) +--R + +--R - cosh(a x)cosh(n log(sinh(a x)) - n log(cosh(a x))) +--R + +--R n + 1 +--R sinh(a x)coth(a x) +--R / +--R (a n + a)sinh(a x) +--R Type: Expression Integer +--E + +--S 17 14:618 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.619~~~~~$\displaystyle @@ -123,7 +298,7 @@ $$ <<*>>= )clear all ---S 5 of 11 +--S 18 aa:=integrate(csch(a*x)^2/coth(a*x),x) --R --R @@ -134,6 +309,43 @@ aa:=integrate(csch(a*x)^2/coth(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 19 +bb:=-1/a*log(coth(a*x)) +--R +--R log(coth(a x)) +--R (2) - -------------- +--R a +--R Type: Expression Integer +--E + +--S 20 +cc:=aa-bb +--R +--R (3) +--R 2cosh(a x) 2sinh(a x) +--R log(coth(a x)) - log(- ---------------------) + log(- ---------------------) +--R sinh(a x) - cosh(a x) sinh(a x) - cosh(a x) +--R ---------------------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 21 +dd:=expandLog cc +--R +--R log(sinh(a x)) + log(coth(a x)) - log(cosh(a x)) +--R (4) ------------------------------------------------ +--R a +--R Type: Expression Integer +--E + +--S 22 14:619 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.620~~~~~$\displaystyle @@ -144,7 +356,7 @@ $$ <<*>>= )clear all ---S 6 of 11 +--S 23 aa:=integrate(1/coth(a*x),x) --R --R @@ -155,6 +367,44 @@ aa:=integrate(1/coth(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=1/a*log(cosh(a*x)) +--R +--R log(cosh(a x)) +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2cosh(a x) +--R - log(cosh(a x)) + log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (3) ----------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 26 +dd:=expandLog cc +--R +--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x +--R (4) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 27 14:620 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R - log(- 1) + log(- 2) +--R (5) --------------------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.621~~~~~$\displaystyle @@ -167,7 +417,7 @@ $$ <<*>>= )clear all ---S 7 of 11 +--S 28 14:621 Axiom cannot compute this integral aa:=integrate(x*coth(a*x),x) --R --R @@ -187,7 +437,7 @@ $$ <<*>>= )clear all ---S 8 of 11 +--S 29 aa:=integrate(x*coth(a*x)^2,x) --R --R @@ -209,6 +459,157 @@ aa:=integrate(x*coth(a*x)^2,x) --R 2a sinh(a x) + 4a cosh(a x)sinh(a x) + 2a cosh(a x) - 2a --R Type: Union(Expression Integer,...) --E + +--S 30 +bb:=x^2/2-(x*coth(a*x)/a)+1/a^2*log(sinh(a*x)) +--R +--R 2 2 +--R 2log(sinh(a x)) - 2a x coth(a x) + a x +--R (2) --------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)log(sinh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 +--R (a x coth(a x) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) - 4a x cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 32 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (a x coth(a x) + log(- 2) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 33 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) + 3) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R (4a x cosh(a x)coth(a x) + (4log(- 2) - 8a x)cosh(a x))sinh(a x) +--R + +--R 2 +--R (a x cosh(2a x) + 2a x cosh(a x) - 3a x)coth(a x) +--R + +--R 2 +--R (log(- 2) - 2a x)cosh(2a x) + (2log(- 2) - 4a x)cosh(a x) - 3log(- 2) +--R + +--R 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a +--R Type: Expression Integer +--E + +--S 35 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +ff:=coshsqrrule ee +--R +--R (8) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 37 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %L sinh(y + x) - %L sinh(y - x) +--I (9) %L cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +gg:=sinhcoshrule ff +--R +--R (10) +--R (- sinh(2a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (a x coth(a x) + log(- 2) - 2a x)sinh(2a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 39 14:622 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R - log(- 1) + log(- 2) +--R (11) --------------------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.623~~~~~$\displaystyle @@ -220,7 +621,7 @@ $$ <<*>>= )clear all ---S 9 of 11 +--S 40 14:623 Axiom cannot compute this integral aa:=integrate(coth(a*x)/x,x) --R --R @@ -240,7 +641,7 @@ $$ <<*>>= )clear all ---S 10 of 11 +--S 41 aa:=integrate(1/(p+q*coth(a*x)),x) --R --R @@ -252,6 +653,54 @@ aa:=integrate(1/(p+q*coth(a*x)),x) --R a q - a p --R Type: Union(Expression Integer,...) --E + +--S 42 +bb:=(p*x)/(p^2-q^2)-q/(a*(p^2-q^2))*log(p*sinh(a*x)+q*cosh(a*x)) +--R +--R q log(p sinh(a x) + q cosh(a x)) - a p x +--R (2) ---------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (3) +--R - 2p sinh(a x) - 2q cosh(a x) +--R - q log(p sinh(a x) + q cosh(a x)) + q log(-----------------------------) +--R sinh(a x) - cosh(a x) +--R + +--R - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 44 +dd:=expandLog cc +--R +--R (4) +--R - q log(p sinh(a x) + q cosh(a x)) - q log(sinh(a x) - cosh(a x)) +--R + +--R q log(- p sinh(a x) - q cosh(a x)) + q log(2) - a q x +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 45 14:624 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R q log(2) - 2q log(- 1) +--R (5) ---------------------- +--R 2 2 +--R a q - a p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.625~~~~~$\displaystyle @@ -262,7 +711,7 @@ $$ <<*>>= )clear all ---S 11 of 11 +--S 46 14:625 Axiom cannot compute this integral aa:=integrate(coth(a*x)^n,x) --R --R diff --git a/src/input/schaum32.input.pamphlet b/src/input/schaum32.input.pamphlet index 085ddf0..e5b2409 100644 --- a/src/input/schaum32.input.pamphlet +++ b/src/input/schaum32.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 10 +--S 1 aa:=integrate(sech(a*x),x) --R --R @@ -27,6 +27,70 @@ aa:=integrate(sech(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=2/a*atan(%e^(a*x)) +--R +--R a x +--R 2atan(%e ) +--R (2) ------------ +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R a x +--R 2atan(sinh(a x) + cosh(a x)) - 2atan(%e ) +--R (3) ------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 5 +dd:=atanrule cc +--R +--R a x +--R - %e + %i - sinh(a x) - cosh(a x) + %i +--R %i log(------------) - %i log(----------------------------) +--R a x sinh(a x) + cosh(a x) + %i +--R %e + %i +--R (5) ----------------------------------------------------------- +--R a +--R Type: Expression Complex Integer +--E + +--S 6 +ee:=expandLog dd +--R +--R (6) +--R %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) - %i) +--R + +--R a x a x +--R - %i log(%e + %i) + %i log(%e - %i) +--R / +--R a +--R Type: Expression Complex Integer +--E + +--S 7 14:626 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.627~~~~~$\displaystyle @@ -37,7 +101,7 @@ $$ <<*>>= )clear all ---S 2 of 10 +--S 8 aa:=integrate(sech(a*x)^2,x) --R --R @@ -47,6 +111,91 @@ aa:=integrate(sech(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=tanh(a*x)/a +--R +--R tanh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2 +--R (3) ------------------------------------------------------------------ +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 11 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 +dd:=sinhsqrrule cc +--R +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) - 1)tanh(a x) - 4 +--R (5) ------------------------------------------------------------------- +--R 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 13 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 14 +ee:=coshsqrrule dd +--R +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)tanh(a x) - 2 +--R (7) ----------------------------------------------------- +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 15 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %L sinh(y + x) - %L sinh(y - x) +--I (8) %L cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 16 +ff:=sinhcoshrule ee +--R +--R (- sinh(2a x) - cosh(2a x) - 1)tanh(a x) - 2 +--R (9) -------------------------------------------- +--R a sinh(2a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 17 14:627 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R 1 +--R (10) - - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.628~~~~~$\displaystyle @@ -57,7 +206,7 @@ $$ <<*>>= )clear all ---S 3 of 10 +--S 18 aa:=integrate(sech(a*x)^3,x) --R --R @@ -83,6 +232,65 @@ aa:=integrate(sech(a*x)^3,x) --R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 19 +bb:=(sech(a*x)*tanh(a*x))/(2*a)+1/(2*a)*atan(sinh(a*x)) +--R +--R atan(sinh(a x)) + sech(a x)tanh(a x) +--R (2) ------------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 20 14:628 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R 2sinh(a x) + 8cosh(a x)sinh(a x) + (12cosh(a x) + 4)sinh(a x) +--R + +--R 3 4 2 +--R (8cosh(a x) + 8cosh(a x))sinh(a x) + 2cosh(a x) + 4cosh(a x) + 2 +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) - 4cosh(a x))sinh(a x) - cosh(a x) - 2cosh(a x) - 1 +--R * +--R atan(sinh(a x)) +--R + +--R 4 3 +--R - sech(a x)sinh(a x) - 4cosh(a x)sech(a x)sinh(a x) +--R + +--R 2 2 +--R (- 6cosh(a x) - 2)sech(a x)sinh(a x) +--R + +--R 3 +--R (- 4cosh(a x) - 4cosh(a x))sech(a x)sinh(a x) +--R + +--R 4 2 +--R (- cosh(a x) - 2cosh(a x) - 1)sech(a x) +--R * +--R tanh(a x) +--R + +--R 3 2 2 +--R 2sinh(a x) + 6cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 +--R 2cosh(a x) - 2cosh(a x) +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) + 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) + 8a cosh(a x))sinh(a x) + 2a cosh(a x) + 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.629~~~~~$\displaystyle @@ -93,7 +301,7 @@ $$ <<*>>= )clear all ---S 4 of 10 +--S 21 aa:=integrate(sech(a*x)^n*tanh(a*x),x) --R --R @@ -111,6 +319,100 @@ aa:=integrate(sech(a*x)^n*tanh(a*x),x) --R a n --R Type: Union(Expression Integer,...) --E + +--S 22 +bb:=-sech(a*x)^n/(n*a) +--R +--R n +--R sech(a x) +--R (2) - ---------- +--R a n +--R Type: Expression Integer +--E + +--S 23 +cc:=aa-bb +--R +--R (3) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R n +--R sech(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 24 +sechrule:=rule(sech(x) == 1/cosh(x)) +--R +--R 1 +--R (4) sech(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +dd:=sechrule cc +--R +--R (5) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R 1 n +--R (---------) +--R cosh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 26 +ee:=expandLog dd +--R +--R (6) +--R sinh +--R 2 2 +--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R - +--R cosh +--R 2 2 +--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R 1 n +--R (---------) +--R cosh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 27 14:629 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.630~~~~~$\displaystyle @@ -121,7 +423,7 @@ $$ <<*>>= )clear all ---S 5 of 10 +--S 28 aa:=integrate(1/sech(a*x),x) --R --R @@ -130,6 +432,22 @@ aa:=integrate(1/sech(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=sinh(a*x)/a +--R +--R sinh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 30 14:630 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.631~~~~~$\displaystyle @@ -142,7 +460,7 @@ $$ <<*>>= )clear all ---S 6 of 10 +--S 31 14:631 Axiom cannot compute this integral aa:=integrate(x*sech(a*x),x) --R --R @@ -162,7 +480,7 @@ $$ <<*>>= )clear all ---S 7 of 10 +--S 32 aa:=integrate(x*sech(a*x)^2,x) --R --R @@ -181,6 +499,161 @@ aa:=integrate(x*sech(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 33 +bb:=(x*tanh(a*x))/a-1/a^2*log(cosh(a*x)) +--R +--R - log(cosh(a x)) + a x tanh(a x) +--R (2) -------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 34 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)log(cosh(a x)) +--R + +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R (- a x sinh(a x) - 2a x cosh(a x)sinh(a x) - a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R 2 2 +--R 2a x sinh(a x) + 4a x cosh(a x)sinh(a x) + 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 35 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 2 +--R (- a x sinh(a x) - 2a x cosh(a x)sinh(a x) - a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R 2 +--R (- log(- 2) + 2a x)sinh(a x) + (- 2log(- 2) + 4a x)cosh(a x)sinh(a x) +--R + +--R 2 +--R (- log(- 2) + 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 36 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (- 4a x cosh(a x)sinh(a x) - a x cosh(2a x) - 2a x cosh(a x) - a x) +--R * +--R tanh(a x) +--R + +--R (- 4log(- 2) + 8a x)cosh(a x)sinh(a x) + (- log(- 2) + 2a x)cosh(2a x) +--R + +--R 2 +--R (- 2log(- 2) + 4a x)cosh(a x) - log(- 2) - 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 38 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 39 +ff:=coshsqrrule ee +--R +--R (8) +--R (2cosh(a x)sinh(a x) + cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (- 2a x cosh(a x)sinh(a x) - a x cosh(2a x) - a x)tanh(a x) +--R + +--R (- 2log(- 2) + 4a x)cosh(a x)sinh(a x) + (- log(- 2) + 2a x)cosh(2a x) +--R + +--R - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 40 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %P sinh(y + x) - %P sinh(y - x) +--I (9) %P cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 41 +gg:=sinhcoshrule ff +--R +--R (10) +--R (sinh(2a x) + cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (- a x sinh(2a x) - a x cosh(2a x) - a x)tanh(a x) +--R + +--R (- log(- 2) + 2a x)sinh(2a x) + (- log(- 2) + 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) + a +--R Type: Expression Integer +--E + +--S 42 14:632 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R log(- 1) - log(- 2) +--R (11) ------------------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.633~~~~~$\displaystyle @@ -192,7 +665,7 @@ $$ <<*>>= )clear all ---S 8 of 10 +--S 43 14:633 Axiom cannot compute this integral aa:=integrate(sech(a*x)/x,x) --R --R @@ -212,7 +685,7 @@ $$ <<*>>= )clear all ---S 9 of 10 +--S 44 aa:=integrate(1/(q+p*sech(a*x)),x) --R --R @@ -259,6 +732,238 @@ aa:=integrate(1/(q+p*sech(a*x)),x) --R a q\|q - p --R Type: Union(List Expression Integer,...) --E + +--S 45 +t1:=integrate(1/(p+q*cosh(a*x)),x) +--R +--R (2) +--R [ +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R , +--R +-------+ +--R | 2 2 +--R (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2atan(-----------------------------------------) +--R 2 2 +--R q - p +--R ------------------------------------------------] +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Union(List Expression Integer,...) +--E + +--S 46 +bb1:=x/q-p/q*t1.1 +--R +--R (3) +--R - +--R p +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R a x\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 47 +bb2:=x/q-p/q*t1.2 +--R +--R +-------+ +--R | 2 2 +-------+ +--R (q sinh(a x) + q cosh(a x) + p)\|q - p | 2 2 +--R - 2p atan(-----------------------------------------) + a x\|q - p +--R 2 2 +--R q - p +--R (4) -------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 48 +cc1:=aa.1-bb1 +--R +--R (5) +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 49 +cc2:=aa.2-bb1 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R p\|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R - 2p\|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 50 +cc3:=aa.1-bb2 +--R +--R (7) +--R +-------+ +--R | 2 2 +--R p\|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2p\|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 51 14:634 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.635~~~~~$\displaystyle @@ -270,7 +975,7 @@ $$ <<*>>= )clear all ---S 10 of 10 +--S 52 14:635 Axiom cannot compute this integral aa:=integrate(sech(a*x)^n,x) --R --R diff --git a/src/input/schaum33.input.pamphlet b/src/input/schaum33.input.pamphlet index c5ee6a2..74e0c8b 100644 --- a/src/input/schaum33.input.pamphlet +++ b/src/input/schaum33.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 10 +--S 1 aa:=integrate(csch(a*x),x) --R --R @@ -27,6 +27,38 @@ aa:=integrate(csch(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/a*log(tanh((a*x)/2)) +--R +--R a x +--R log(tanh(---)) +--R 2 +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R (3) +--R a x +--R - log(tanh(---)) - log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R log(sinh(a x) + cosh(a x) - 1) +--R / +--R a +--R Type: Expression Integer +--E + +--S 4 14:636 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.637~~~~~$\displaystyle @@ -37,7 +69,7 @@ $$ <<*>>= )clear all ---S 2 of 10 +--S 5 aa:=integrate(csch(a*x)^2,x) --R --R @@ -47,6 +79,30 @@ aa:=integrate(csch(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 6 +bb:=-coth(a*x)/a +--R +--R coth(a x) +--R (2) - --------- +--R a +--R Type: Expression Integer +--E + +--S 7 14:637 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 +--R coth(a x)sinh(a x) + 2cosh(a x)coth(a x)sinh(a x) +--R + +--R 2 +--R (cosh(a x) - 1)coth(a x) - 2 +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.638~~~~~$\displaystyle @@ -57,7 +113,7 @@ $$ <<*>>= )clear all ---S 3 of 10 +--S 8 aa:=integrate(csch(a*x)^3,x) --R --R @@ -93,6 +149,72 @@ aa:=integrate(csch(a*x)^3,x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=-(csch(a*x)*coth(a*x))/(2*a)-1/(2*a)*log(tanh((a*x)/2)) +--R +--R a x +--R - log(tanh(---)) - coth(a x)csch(a x) +--R 2 +--R (2) ------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 10 14:638 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R a x +--R log(tanh(---)) +--R 2 +--R + +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 4 +--R coth(a x)csch(a x)sinh(a x) +--R + +--R 3 +--R (4cosh(a x)coth(a x)csch(a x) - 2)sinh(a x) +--R + +--R 2 2 +--R ((6cosh(a x) - 2)coth(a x)csch(a x) - 6cosh(a x))sinh(a x) +--R + +--R 3 2 +--R ((4cosh(a x) - 4cosh(a x))coth(a x)csch(a x) - 6cosh(a x) - 2)sinh(a x) +--R + +--R 4 2 3 +--R (cosh(a x) - 2cosh(a x) + 1)coth(a x)csch(a x) - 2cosh(a x) - 2cosh(a x) +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a x) +--R + +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.639~~~~~$\displaystyle @@ -103,7 +225,7 @@ $$ <<*>>= )clear all ---S 4 of 10 +--S 11 aa:=integrate(csch(a*x)^n*coth(a*x),x) --R --R @@ -121,6 +243,202 @@ aa:=integrate(csch(a*x)^n*coth(a*x),x) --R a n --R Type: Union(Expression Integer,...) --E + +--S 12 +bb:=-csch(a*x)^n/(n*a) +--R +--R n +--R csch(a x) +--R (2) - ---------- +--R a n +--R Type: Expression Integer +--E + +--S 13 +cc:=aa-bb +--R +--R (3) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R n +--R csch(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 14 +cschrule:=rule(csch(x) == 1/sinh(x)) +--R +--R 1 +--R (4) csch(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 15 +dd:=cschrule cc +--R +--R (5) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 16 +ee:=expandLog dd +--R +--R (6) +--R sinh +--R 2 2 +--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R - +--R cosh +--R 2 2 +--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 17 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (7) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 18 +ff:=sinhsqrrule ee +--R +--R (8) +--R sinh +--R 2 +--R 4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) - 3 +--R n log(--------------------------------------------------) +--R 2 +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R - +--R cosh +--R 2 +--R 4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) - 3 +--R n log(--------------------------------------------------) +--R 2 +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 19 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (9) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 +gg:=coshsqrrule ff +--R +--R (10) +--R sinh +--R n log(2cosh(a x)sinh(a x) + cosh(2a x) - 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R - +--R cosh +--R n log(2cosh(a x)sinh(a x) + cosh(2a x) - 1) +--R + +--R - n log(sinh(a x) + cosh(a x)) - n log(2) +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 21 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %O sinh(y + x) - %O sinh(y - x) +--I (11) %O cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 22 +hh:=sinhcoshrule gg +--R +--R (12) +--R sinh +--R n log(sinh(2a x) + cosh(2a x) - 1) - n log(sinh(a x) + cosh(a x)) +--R + +--R - n log(2) +--R + +--R - +--R cosh +--R n log(sinh(2a x) + cosh(2a x) - 1) - n log(sinh(a x) + cosh(a x)) +--R + +--R - n log(2) +--R + +--R 1 n +--R (---------) +--R sinh(a x) +--R / +--R a n +--R Type: Expression Integer +--E + +--S 23 14:639 Schaums and Axiom agree +ii:=complexNormalize hh +--R +--R (13) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.640~~~~~$\displaystyle @@ -131,7 +449,7 @@ $$ <<*>>= )clear all ---S 5 of 10 +--S 24 aa:=integrate(1/csch(a*x),x) --R --R @@ -140,6 +458,22 @@ aa:=integrate(1/csch(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 25 +bb:=1/a*cosh(a*x) +--R +--R cosh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 26 14:640 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.641~~~~~$\displaystyle @@ -152,7 +486,7 @@ $$ <<*>>= )clear all ---S 6 of 10 +--S 27 14:641 Axiom cannot compute this integral aa:=integrate(x*csch(a*x),x) --R --R @@ -172,7 +506,7 @@ $$ <<*>>= )clear all ---S 7 of 10 +--S 28 aa:=integrate(x*csch(a*x)^2,x) --R --R @@ -191,6 +525,156 @@ aa:=integrate(x*csch(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=-(x*coth(a*x))/a+1/a^2*log(sinh(a*x)) +--R +--R log(sinh(a x)) - a x coth(a x) +--R (2) ------------------------------ +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 30 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)log(sinh(a x)) +--R + +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 +--R (a x coth(a x) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) - 4a x cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 31 +dd:=expandLog cc +--R +--R (4) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R 2 +--R (a x coth(a x) + log(- 2) - 2a x)sinh(a x) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R 2 2 +--R (a x cosh(a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(a x) - log(- 2) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 32 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 33 +ee:=sinhsqrrule dd +--R +--R (6) +--R 2 +--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) + 3) +--R * +--R log(sinh(a x) - cosh(a x)) +--R + +--R (4a x cosh(a x)coth(a x) + (4log(- 2) - 8a x)cosh(a x))sinh(a x) +--R + +--R 2 +--R (a x cosh(2a x) + 2a x cosh(a x) - 3a x)coth(a x) +--R + +--R 2 +--R (log(- 2) - 2a x)cosh(2a x) + (2log(- 2) - 4a x)cosh(a x) - 3log(- 2) +--R + +--R 2a x +--R / +--R 2 2 2 2 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a +--R Type: Expression Integer +--E + +--S 34 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 +ff:=coshsqrrule ee +--R +--R (8) +--R (- 2cosh(a x)sinh(a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 36 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %P sinh(y + x) - %P sinh(y - x) +--I (9) %P cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +gg:=sinhcoshrule ff +--R +--R (10) +--R (- sinh(2a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x)) +--R + +--R (a x coth(a x) + log(- 2) - 2a x)sinh(2a x) +--R + +--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) - log(- 2) +--R / +--R 2 2 2 +--R a sinh(2a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 38 14:642 Axiom cannot simplify this expression +hh:=complexNormalize gg +--R +--R - log(- 1) + log(- 2) +--R (11) --------------------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.643~~~~~$\displaystyle @@ -202,7 +686,7 @@ $$ <<*>>= )clear all ---S 8 of 10 +--S 39 14:643 Axiom cannot compute this integral aa:=integrate(csch(a*x)/x,x) --R --R @@ -222,7 +706,7 @@ $$ <<*>>= )clear all ---S 9 of 10 +--S 40 aa:=integrate(1/(q+p*csch(a*x)),x) --R --R @@ -257,6 +741,251 @@ aa:=integrate(1/(q+p*csch(a*x)),x) --R a q\|q + p --R Type: Union(Expression Integer,...) --E + +--S 41 +t1:=integrate(1/(p+q*sinh(a*x)),x) +--R +--R (2) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=x/q-p/q*t1 +--R +--R (3) +--R - +--R p +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 +--R a x\|q + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R (4) +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 44 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (5) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 45 +dd:=sinhsqrrule cc +--R +--R (6) +--R p +--R * +--R log +--R 2 2 +--R (4q cosh(a x) + 4p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 2 2 +--R 2q cosh(a x) + 4p q cosh(a x) + q + 4p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (4q + 4p q)sinh(a x) + (4q + 4p q)cosh(a x) + 4p q + 4p +--R / +--R 2 +--R (4q cosh(a x) + 4p)sinh(a x) + q cosh(2a x) + 2q cosh(a x) +--R + +--R 4p cosh(a x) - 3q +--R + +--R p +--R * +--R log +--R 2 2 +--R (4q cosh(a x) + 4p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 2 2 +--R 2q cosh(a x) + 4p q cosh(a x) + q + 4p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 4q - 4p q)sinh(a x) + (- 4q - 4p q)cosh(a x) - 4p q - 4p +--R / +--R 2 +--R (4q cosh(a x) + 4p)sinh(a x) + q cosh(2a x) + 2q cosh(a x) +--R + +--R 4p cosh(a x) - 3q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 46 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (7) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 47 +ee:=coshsqrrule dd +--R +--R (8) +--R p +--R * +--R log +--R 2 2 +--R (2q cosh(a x) + 2p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p +--R / +--R (2q cosh(a x) + 2p)sinh(a x) + q cosh(2a x) + 2p cosh(a x) - q +--R + +--R p +--R * +--R log +--R 2 2 +--R (2q cosh(a x) + 2p q)sinh(a x) + q cosh(2a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R (2q cosh(a x) + 2p)sinh(a x) + q cosh(2a x) + 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E + +--S 48 14:644 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R 4 2 2 +--R p log(q + p q ) +--R (9) ---------------- +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.645~~~~~$\displaystyle @@ -268,7 +997,7 @@ $$ <<*>>= )clear all ---S 10 of 10 +--S 49 14:645 Axiom cannot compute this integral aa:=integrate(csch(a*x)^n,x) --R --R diff --git a/src/input/schaum34.input.pamphlet b/src/input/schaum34.input.pamphlet index 1a5c359..83d7061 100644 --- a/src/input/schaum34.input.pamphlet +++ b/src/input/schaum34.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 32 +--S 1 aa:=integrate(asinh(x/a),x) --R --R @@ -33,6 +33,69 @@ aa:=integrate(asinh(x/a),x) --R \|x + a - x --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=x*asinh(x/a)-sqrt(x^2+a^2) +--R +--R +-------+ +--R | 2 2 x +--R (2) - \|x + a + x asinh(-) +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x + a + x x +--R (3) x log(--------------) - x asinh(-) +--R a a +--R Type: Expression Integer +--E + +--S 4 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R \|x + a + x \| a +--R (5) x log(--------------) - x log(---------------) +--R a a +--R Type: Expression Integer +--E + +--S 6 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 |x + a +--R (6) x log(\|x + a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R Type: Expression Integer +--E + +--S 7 14:646 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.647~~~~~$\displaystyle @@ -44,7 +107,7 @@ $$ <<*>>= )clear all ---S 2 of 32 +--S 8 aa:=integrate(x*asinh(x/a),x) --R --R @@ -64,6 +127,77 @@ aa:=integrate(x*asinh(x/a),x) --R 8x\|x + a - 8x - 4a --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=(x^2/2+a^2/4)*asinh(x/a)-(x*sqrt(x^2+a^2))/4 +--R +--R +-------+ +--R | 2 2 2 2 x +--R - x\|x + a + (2x + a )asinh(-) +--R a +--R (2) ---------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 2 2 \|x + a + x 2 2 x +--R (2x + a )log(--------------) + (- 2x - a )asinh(-) +--R a a +--R (3) ---------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 11 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 2 2 \|x + a + x 2 2 \| a +--R (2x + a )log(--------------) + (- 2x - a )log(---------------) +--R a a +--R (5) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 13 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R 2 2 | 2 2 2 2 |x + a +--R (2x + a )log(\|x + a + x) + (- 2x - a )log(a |------- + x) +--R | 2 +--R \| a +--R (6) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 14 14:647 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.648~~~~~$\displaystyle @@ -74,7 +208,7 @@ $$ <<*>>= )clear all ---S 3 of 32 +--S 15 aa:=integrate(x^2*asinh(x/a),x) --R --R @@ -94,6 +228,77 @@ aa:=integrate(x^2*asinh(x/a),x) --R (36x + 9a )\|x + a - 36x - 27a x --R Type: Union(Expression Integer,...) --E + +--S 16 +bb:=x^3/3*asinh(x/a)+((2*a^2-x^2)*sqrt(x^2+a^2))/9 +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (- x + 2a )\|x + a + 3x asinh(-) +--R a +--R (2) ------------------------------------ +--R 9 +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 3 \|x + a + x 3 x +--R x log(--------------) - x asinh(-) +--R a a +--R (3) ---------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 18 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 +dd:=asinhlogrule cc +--R +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 3 \|x + a + x 3 \| a +--R x log(--------------) - x log(---------------) +--R a a +--R (5) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 20 +ee:=expandLog dd +--R +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 3 |x + a +--R x log(\|x + a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R (6) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 21 14:648 Schaums and Axiom agree +ff:=rootSimp ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.649~~~~~$\displaystyle @@ -124,7 +329,7 @@ $$ <<*>>= )clear all ---S 4 of 32 +--S 22 14:649 Axiom cannot compute this integral aa:=integrate(asinh(x/a)/x,x) --R --R @@ -146,7 +351,7 @@ $$ <<*>>= )clear all ---S 5 of 32 +--S 23 aa:=integrate(asinh(x/a)/x^2,x) --R --R @@ -164,6 +369,113 @@ aa:=integrate(asinh(x/a)/x^2,x) --R a x --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=-asinh(x/a)/x-1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a x +--R - x log(--------------) - a asinh(-) +--R x a +--R (2) ------------------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R (3) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x + a + x \|x + a + a x +--R - a log(--------------) + x log(--------------) + a asinh(-) +--R a x a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 26 +asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (4) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 +dd:=asinhlogrule cc +--R +--R (5) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R |x + a +--R +-------+ +-------+ a |------- + x +--R | 2 2 | 2 2 | 2 +--R \|x + a + x \|x + a + a \| a +--R - a log(--------------) + x log(--------------) + a log(---------------) +--R a x a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 28 +ee:=expandLog dd +--R +--R (6) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - a log(\|x + a + x) + x log(\|x + a + a) +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R |x + a +--R a log(a |------- + x) - x log(x) +--R | 2 +--R \| a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 29 +ff:=rootSimp ee +--R +--R (7) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) +--R + +--R - log(x) +--R / +--R a +--R Type: Expression Integer +--E + +--S 30 14:650 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R log(- 1) +--R (8) - -------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.651~~~~~$\displaystyle @@ -182,7 +494,7 @@ $$ <<*>>= )clear all ---S 6 of 32 +--S 31 aa:=integrate(acosh(x/a),x) --R --R @@ -197,6 +509,100 @@ aa:=integrate(acosh(x/a),x) --R \|x - a - x --R Type: Union(Expression Integer,...) --E + +--S 32 +bb1:=x*acosh(x/a)-sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 x +--R (2) - \|x - a + x acosh(-) +--R a +--R Type: Expression Integer +--E + +--S 33 +bb2:=x*acosh(x/a)+sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 x +--R (3) \|x - a + x acosh(-) +--R a +--R Type: Expression Integer +--E + +--S 34 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R \|x - a + x x +--R (4) x log(--------------) - x acosh(-) +--R a a +--R Type: Expression Integer +--E + +--S 35 +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 \|x - a + x x | 2 2 +--R (x\|x - a - x )log(--------------) + (- x acosh(-) + 2x)\|x - a +--R a a +--R + +--R 2 x 2 2 +--R x acosh(-) - 2x + 2a +--R a +--R / +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Expression Integer +--E + +--S 36 +acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (6) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +dd1:=acoshlogrule cc1 +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R \|x - a + x \| a +--R (7) x log(--------------) - x log(---------------) +--R a a +--R Type: Expression Integer +--E + +--S 38 +ee1:=expandLog dd1 +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 |x - a +--R (8) x log(\|x - a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R Type: Expression Integer +--E + +--S 39 14:651 Schaums and Axiom agree +ff1:=rootSimp ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.652~~~~~$\displaystyle @@ -217,7 +623,7 @@ $$ <<*>>= )clear all ---S 7 of 32 +--S 40 aa:=integrate(x*acosh(x/a),x) --R --R @@ -237,6 +643,114 @@ aa:=integrate(x*acosh(x/a),x) --R 8x\|x - a - 8x + 4a --R Type: Union(Expression Integer,...) --E + +--S 41 +bb1:=1/4*(2*x^2-a^2)*acosh(x/a)-1/4*x*sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 2 2 x +--R - x\|x - a + (2x - a )acosh(-) +--R a +--R (2) ---------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 42 +bb2:=1/4*(2*x^2-a^2)*acosh(x/a)+1/4*x*sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 2 2 x +--R x\|x - a + (2x - a )acosh(-) +--R a +--R (3) -------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 43 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 2 2 \|x - a + x 2 2 x +--R (2x - a )log(--------------) + (- 2x + a )acosh(-) +--R a a +--R (4) ---------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 44 +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 | 2 2 4 2 2 4 \|x - a + x +--R ((4x - 2a x)\|x - a - 4x + 4a x - a )log(--------------) +--R a +--R + +--R +-------+ +--R 3 2 x 3 2 | 2 2 +--R ((- 4x + 2a x)acosh(-) + 4x - 2a x)\|x - a +--R a +--R + +--R 4 2 2 4 x 4 2 2 +--R (4x - 4a x + a )acosh(-) - 4x + 4a x +--R a +--R / +--R +-------+ +--R | 2 2 2 2 +--R 8x\|x - a - 8x + 4a +--R Type: Expression Integer +--E + +--S 45 +acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (6) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 46 +dd1:=acoshlogrule cc1 +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 2 2 \|x - a + x 2 2 \| a +--R (2x - a )log(--------------) + (- 2x + a )log(---------------) +--R a a +--R (7) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 47 +ee1:=expandLog dd1 +--R +--R +-------+ +--R +-------+ | 2 2 +--R 2 2 | 2 2 2 2 |x - a +--R (2x - a )log(\|x - a + x) + (- 2x + a )log(a |------- + x) +--R | 2 +--R \| a +--R (8) ---------------------------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 48 14:652 Schaums and Axiom agree +ff1:=rootSimp ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.653~~~~~$\displaystyle @@ -257,7 +771,7 @@ $$ <<*>>= )clear all ---S 8 of 32 +--S 49 aa:=integrate(x^2*acosh(x/a),x) --R --R @@ -277,6 +791,114 @@ aa:=integrate(x^2*acosh(x/a),x) --R (36x - 9a )\|x - a - 36x + 27a x --R Type: Union(Expression Integer,...) --E + +--S 50 +bb1:=1/3*x^3*acosh(x/a)-1/9*(x^2+2*a^2)*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (- x - 2a )\|x - a + 3x acosh(-) +--R a +--R (2) ------------------------------------ +--R 9 +--R Type: Expression Integer +--E + +--S 51 +bb2:=1/3*x^3*acosh(x/a)+1/9*(x^2+2*a^2)*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (x + 2a )\|x - a + 3x acosh(-) +--R a +--R (3) ---------------------------------- +--R 9 +--R Type: Expression Integer +--E + +--S 52 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 3 \|x - a + x 3 x +--R x log(--------------) - x acosh(-) +--R a a +--R (4) ---------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 53 +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 \|x - a + x +--R ((12x - 3a x )\|x - a - 12x + 9a x )log(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 x 5 2 3 4 | 2 2 +--R ((- 12x + 3a x )acosh(-) + 8x + 10a x - 12a x)\|x - a +--R a +--R + +--R 6 2 4 x 6 2 4 4 2 6 +--R (12x - 9a x )acosh(-) - 8x - 6a x + 18a x - 4a +--R a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (36x - 9a )\|x - a - 36x + 27a x +--R Type: Expression Integer +--E + +--S 54 +acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (6) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 55 +dd1:=acoshlogrule cc1 +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R +-------+ a |------- + x +--R | 2 2 | 2 +--R 3 \|x - a + x 3 \| a +--R x log(--------------) - x log(---------------) +--R a a +--R (7) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 56 +ee1:=expandLog dd1 +--R +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 3 |x - a +--R x log(\|x - a + x) - x log(a |------- + x) +--R | 2 +--R \| a +--R (8) ---------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 57 14:653 Schaums and Axiom agree +ff1:=rootSimp ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.654~~~~~$\displaystyle @@ -296,7 +918,7 @@ $$ <<*>>= )clear all ---S 9 of 32 +--S 58 14:654 Axiom cannot compute this integral aa:=integrate(acosh(x/a)/x,x) --R --R @@ -325,7 +947,7 @@ $$ <<*>>= )clear all ---S 10 of 32 +--S 59 aa:=integrate(acosh(x/a)/x^2,x) --R --R @@ -338,6 +960,68 @@ aa:=integrate(acosh(x/a)/x^2,x) --R a x --R Type: Union(Expression Integer,...) --E + +--S 60 +bb1:=-acosh(x/a)/x-1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a x +--R - x log(--------------) - a acosh(-) +--R x a +--R (2) ------------------------------------ +--R a x +--R Type: Expression Integer +--E + +--S 61 +bb2:=-acosh(x/a)/x+1/a*log((a+sqrt(x^2+a^2))/x) +--R +--R +-------+ +--R | 2 2 +--R \|x + a + a x +--R x log(--------------) - a acosh(-) +--R x a +--R (3) ---------------------------------- +--R a x +--R Type: Expression Integer +--E + +--S 62 +cc1:=aa-bb1 +--R +--R (4) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R \|x + a + a \|x - a + x \|x - a - x +--R x log(--------------) - a log(--------------) + 2x atan(--------------) +--R x a a +--R + +--R x +--R a acosh(-) +--R a +--R / +--R a x +--R Type: Expression Integer +--E + +--S 63 14:655 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R \|x + a + a \|x - a + x \|x - a - x +--R - x log(--------------) - a log(--------------) + 2x atan(--------------) +--R x a a +--R + +--R x +--R a acosh(-) +--R a +--R / +--R a x +--R Type: Expression Integer +--E @ \section{\cite{1}:14.656~~~~~$\displaystyle @@ -348,7 +1032,7 @@ $$ <<*>>= )clear all ---S 11 of 32 +--S 64 aa:=integrate(atanh(x/a),x) --R --R @@ -359,6 +1043,58 @@ aa:=integrate(atanh(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 65 +bb:=x*atanh(x/a)+a/2*log(a^2-x^2) +--R +--R 2 2 x +--R a log(- x + a ) + 2x atanh(-) +--R a +--R (2) ------------------------------ +--R 2 +--R Type: Expression Integer +--E + +--S 66 +cc:=aa-bb +--R +--R 2 2 - x - a 2 2 x +--R a log(x - a ) + x log(-------) - a log(- x + a ) - 2x atanh(-) +--R x - a a +--R (3) ---------------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 67 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 68 +dd:=atanhrule cc +--R +--R 2 2 2 2 +--R a log(x - a ) - a log(- x + a ) +--R (5) --------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 69 14:656 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R a log(- 1) +--R (6) ---------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.657~~~~~$\displaystyle @@ -369,7 +1105,7 @@ $$ <<*>>= )clear all ---S 12 of 32 +--S 70 aa:=integrate(x*atanh(x/a),x) --R --R @@ -380,6 +1116,46 @@ aa:=integrate(x*atanh(x/a),x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 71 +bb:=(a*x)/2+1/2*(x^2-a^2)*atanh(x/a) +--R +--R 2 2 x +--R (x - a )atanh(-) + a x +--R a +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 72 +cc:=aa-bb +--R +--R 2 2 - x - a 2 2 x +--R (x - a )log(-------) + (- 2x + 2a )atanh(-) +--R x - a a +--R (3) --------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 73 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 74 14:657 Schaums and Axiom agree +dd:=atanhrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.658~~~~~$\displaystyle @@ -391,7 +1167,7 @@ $$ <<*>>= )clear all ---S 13 of 32 +--S 75 aa:=integrate(x^2*atanh(x/a),x) --R --R @@ -402,6 +1178,59 @@ aa:=integrate(x^2*atanh(x/a),x) --R 6 --R Type: Union(Expression Integer,...) --E + +--S 76 +bb:=(a*x^2)/6+x^3/3*atanh(x/a)+a^3/6*log(a^2-x^2) +--R +--R 3 2 2 3 x 2 +--R a log(- x + a ) + 2x atanh(-) + a x +--R a +--R (2) ------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 77 +cc:=aa-bb +--R +--R 3 2 2 3 - x - a 3 2 2 3 x +--R a log(x - a ) + x log(-------) - a log(- x + a ) - 2x atanh(-) +--R x - a a +--R (3) ---------------------------------------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 78 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 79 +dd:=atanhrule cc +--R +--R 3 2 2 3 2 2 +--R a log(x - a ) - a log(- x + a ) +--R (5) --------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 80 14:658 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3 +--R a log(- 1) +--R (6) ---------- +--R 6 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.659~~~~~$\displaystyle @@ -412,7 +1241,7 @@ $$ <<*>>= )clear all ---S 14 of 32 +--S 81 14:659 Axiom cannot compute this integral aa:=integrate(atanh(x/a)/x,x) --R --R @@ -433,7 +1262,7 @@ $$ <<*>>= )clear all ---S 15 of 32 +--S 82 aa:=integrate(atanh(x/a)/x^2,x) --R --R @@ -444,6 +1273,70 @@ aa:=integrate(atanh(x/a)/x^2,x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 83 +bb:=-atanh(x/a)/x+1/(2*a)*log(x^2/(a^2-x^2)) +--R +--R 2 +--R x x +--R x log(- -------) - 2a atanh(-) +--R 2 2 a +--R x - a +--R (2) ------------------------------ +--R 2a x +--R Type: Expression Integer +--E + +--S 84 +cc:=aa-bb +--R +--R (3) +--R 2 +--R 2 2 x - x - a +--R - x log(x - a ) + 2x log(x) - x log(- -------) - a log(-------) +--R 2 2 x - a +--R x - a +--R + +--R x +--R 2a atanh(-) +--R a +--R / +--R 2a x +--R Type: Expression Integer +--E + +--S 85 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (4) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 86 +dd:=atanhrule cc +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(- -------) +--R 2 2 +--R x - a +--R (5) ----------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 87 14:660 Schaums and Axiom agree +ee:=expandLog dd +--R +--R log(- 1) +--R (6) - -------- +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.661~~~~~$\displaystyle @@ -451,10 +1344,15 @@ aa:=integrate(atanh(x/a)/x^2,x) $$\int{\coth^{-1}\frac{x}{a}}= x\coth^{-1}{x}+\frac{a}{2}\ln(x^2-a^2) $$ + +Note that it appears there is a typo in Schaums (1968 printing 4). +$$\int{\coth^{-1}\frac{x}{a}}= +x\coth^{-1}{x/a}+\frac{a}{2}\ln(x^2-a^2) +$$ <<*>>= )clear all ---S 16 of 32 +--S 88 aa:=integrate(acoth(x/a),x) --R --R @@ -465,6 +1363,46 @@ aa:=integrate(acoth(x/a),x) --R 2 --R Type: Union(Expression Integer,...) --E + +--S 89 +bb:=x*acoth(x/a)+a/2*log(x^2-a^2) +--R +--R 2 2 x +--R a log(x - a ) + 2x acoth(-) +--R a +--R (2) ---------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 90 +cc:=aa-bb +--R +--R x + a x +--R x log(-----) - 2x acoth(-) +--R x - a a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 91 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 92 14:661 Schaums and Axiom agree +dd:=acothrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.662~~~~~$\displaystyle @@ -475,7 +1413,7 @@ $$ <<*>>= )clear all ---S 17 of 32 +--S 93 aa:=integrate(x*acoth(x/a),x) --R --R @@ -486,6 +1424,46 @@ aa:=integrate(x*acoth(x/a),x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 94 +bb:=(a*x)/2+1/2*(x^2-a^2)*acoth(x/a) +--R +--R 2 2 x +--R (x - a )acoth(-) + a x +--R a +--R (2) ----------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 95 +cc:=aa-bb +--R +--R 2 2 x + a 2 2 x +--R (x - a )log(-----) + (- 2x + 2a )acoth(-) +--R x - a a +--R (3) ------------------------------------------- +--R 4 +--R Type: Expression Integer +--E + +--S 96 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 97 14:662 Schaums and Axiom agree +dd:=acothrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.663~~~~~$\displaystyle @@ -497,7 +1475,7 @@ $$ <<*>>= )clear all ---S 18 of 32 +--S 98 aa:=integrate(x^2*acoth(x/a),x) --R --R @@ -508,6 +1486,46 @@ aa:=integrate(x^2*acoth(x/a),x) --R 6 --R Type: Union(Expression Integer,...) --E + +--S 99 +bb:=(a*x^2)/6+x^3/3*acoth(x/a)+a^3/6*log(x^2-a^2) +--R +--R 3 2 2 3 x 2 +--R a log(x - a ) + 2x acoth(-) + a x +--R a +--R (2) ----------------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 100 +cc:=aa-bb +--R +--R 3 x + a 3 x +--R x log(-----) - 2x acoth(-) +--R x - a a +--R (3) -------------------------- +--R 6 +--R Type: Expression Integer +--E + +--S 101 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 102 14:663 Schaums and Axiom agree +dd:=acothrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.664~~~~~$\displaystyle @@ -518,7 +1536,7 @@ $$ <<*>>= )clear all ---S 19 of 32 +--S 103 14:664 Axiom cannot compute this integral aa:=integrate(acoth(x/a)/x,x) --R --R @@ -539,7 +1557,7 @@ $$ <<*>>= )clear all ---S 20 of 32 +--S 104 aa:=integrate(acoth(x/a)/x^2,x) --R --R @@ -550,6 +1568,64 @@ aa:=integrate(acoth(x/a)/x^2,x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 105 +bb:=-acoth(x/a)/x+1/(2*a)*log(x^2/(x^2-a^2)) +--R +--R 2 +--R x x +--R x log(-------) - 2a acoth(-) +--R 2 2 a +--R x - a +--R (2) ---------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 106 +cc:=aa-bb +--R +--R (3) +--R 2 +--R 2 2 x + a x x +--R - x log(x - a ) + 2x log(x) - a log(-----) - x log(-------) + 2a acoth(-) +--R x - a 2 2 a +--R x - a +--R -------------------------------------------------------------------------- +--R 2a x +--R Type: Expression Integer +--E + +--S 107 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (4) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 108 +dd:=acothrule cc +--R +--R 2 +--R 2 2 x +--R - log(x - a ) + 2log(x) - log(-------) +--R 2 2 +--R x - a +--R (5) --------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 109 14:665 Schaums and Axiom agree +ee:=expandLog dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.666~~~~~$\displaystyle @@ -568,7 +1644,7 @@ $$ <<*>>= )clear all ---S 21 of 32 +--S 110 aa:=integrate(asech(x/a),x) --R --R @@ -579,7 +1655,308 @@ aa:=integrate(asech(x/a),x) --R x x --R Type: Union(Expression Integer,...) --E + +--S 111 +bb1:=x*asech(x/a)+a*asin(x/a) +--R +--R x x +--R (2) a asin(-) + x asech(-) +--R a a +--R Type: Expression Integer +--E + +--S 112 +bb2:=x*asech(x/a)-a*asin(x/a) +--R +--R x x +--R (3) - a asin(-) + x asech(-) +--R a a +--R Type: Expression Integer +--E + +--S 113 +cc1:=aa-bb1 +--R +--R (4) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a x x +--R x log(----------------) - 2a atan(----------------) - a asin(-) - x asech(-) +--R x x a a +--R Type: Expression Integer +--E + +--S 114 +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a x x +--R x log(----------------) - 2a atan(----------------) + a asin(-) - x asech(-) +--R x x a a +--R Type: Expression Integer +--E + +--S 115 +asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1))) +--R +--R +--------+ +--R | 2 +--R |- x + 1 +--R x |-------- + 1 +--R | 2 +--R \| x +--R (6) asech(x) == log(----------------) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 116 +dd1:=asechrule cc1 +--R +--R (7) +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R \| x \|- x + a + a +--R - x log(-----------------) + x log(----------------) +--R x x +--R + +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 117 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (8) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 118 +ee1:=asinrule dd1 +--R +--R (9) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) - %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R x log(----------------) - 2a atan(----------------) +--R x x +--R Type: Expression Complex Integer +--E + +--S 119 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (10) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 120 +ff1:=atanrule ee1 +--R +--R (11) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) - %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a - \|- x + a + %i x + a +--R x log(----------------) + %i a log(-------------------------) +--R x +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 121 +gg1:=expandLog ff1 +--R +--R (12) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R - x log(x |--------- + a) - %i a log(a |--------- - %i x) +--R | 2 | 2 +--R \| x \| a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - %i a log(\|- x + a + %i x - a) + x log(\|- x + a + a) +--R + +--R +---------+ +--R | 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 122 +hh1:=rootSimp gg1 +--R +--R (13) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 123 14:666 Schaums and Axiom agree +ii1:=complexNormalize hh1 +--R +--R (14) 0 +--R Type: Expression Complex Integer +--E + +@ +Note that Axiom has a built-in assumption about the sign of asech(x/a). +We can see this if we simplify the cc2 value and show that it differs +by a complex value of x. +<<*>>= +--S 124 +dd2:=asechrule cc2 +--R +--R (15) +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R \| x \|- x + a + a +--R - x log(-----------------) + x log(----------------) +--R x x +--R + +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R - 2a atan(----------------) + a asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 125 +ee2:=asinrule dd2 +--R +--R (16) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) + %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R x log(----------------) - 2a atan(----------------) +--R x x +--R Type: Expression Complex Integer +--E + +--S 126 +ff2:=atanrule ee2 +--R +--R (17) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R x |--------- + a a |--------- - %i x +--R | 2 | 2 +--R \| x \| a +--R - x log(-----------------) + %i a log(--------------------) +--R x a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a - \|- x + a + %i x + a +--R x log(----------------) + %i a log(-------------------------) +--R x +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 127 +gg2:=expandLog ff2 +--R +--R (18) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R |- x + a |- x + a +--R - x log(x |--------- + a) + %i a log(a |--------- - %i x) +--R | 2 | 2 +--R \| x \| a +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - %i a log(\|- x + a + %i x - a) + x log(\|- x + a + a) +--R + +--R +---------+ +--R | 2 2 +--R %i a log(\|- x + a - %i x - a) - %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 128 +hh2:=rootSimp gg2 +--R +--R (19) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) + %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R %i a log(%i\|x - a - %i x - a) - %i a log(a) + %i a log(- 1) +--R Type: Expression Complex Integer +--E + +--S 129 +ii2:=complexNormalize hh2 +--R +--R +-------+ +--R | 2 2 +--R (20) 2%i a log(%i\|x - a - %i x) - 2%i a log(a) +--R Type: Expression Complex Integer +--E + @ +Thus we can conjecture that solutions that show up with x in only the +imaginary part do so when the assumption of the sign of an inverse +function differs. \section{\cite{1}:14.667~~~~~$\displaystyle \int{x{\rm ~sech}^{-1}\frac{x}{a}}~dx$} @@ -599,7 +1976,7 @@ $$ <<*>>= )clear all ---S 22 of 32 +--S 130 aa:=integrate(x*asech(x/a),x) --R --R @@ -614,6 +1991,118 @@ aa:=integrate(x*asech(x/a),x) --R 2\|- x + a - 2a --R Type: Union(Expression Integer,...) --E + +--S 131 +bb1:=1/2*x^2*asech(x/a)-1/2*a*sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 2 x +--R - a\|- x + a + x asech(-) +--R a +--R (2) ---------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 132 +bb2:=1/2*x^2*asech(x/a)+1/2*a*sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 2 x +--R a\|- x + a + x asech(-) +--R a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 133 +cc1:=aa-bb1 +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a + a 2 x 2 +--R x log(----------------) - x asech(-) - a +--R x a +--R (4) ----------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 134 +cc2:=aa-bb2 +--R +--R (5) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 2 \|- x + a + a +--R (x \|- x + a - a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 x 2 | 2 2 2 x 2 3 +--R (- x asech(-) + a )\|- x + a + a x asech(-) + 2a x - a +--R a a +--R / +--R +---------+ +--R | 2 2 +--R 2\|- x + a - 2a +--R Type: Expression Integer +--E + +--S 135 +asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1))) +--R +--R +--------+ +--R | 2 +--R |- x + 1 +--R x |-------- + 1 +--R | 2 +--R \| x +--R (6) asech(x) == log(----------------) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 136 +dd1:=asechrule cc1 +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R x |--------- + a +---------+ +--R | 2 | 2 2 +--R 2 \| x 2 \|- x + a + a 2 +--R - x log(-----------------) + x log(----------------) - a +--R x x +--R (7) --------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 137 +ee1:=expandLog dd1 +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 2 +--R - x log(x |--------- + a) + x log(\|- x + a + a) - a +--R | 2 +--R \| x +--R (8) --------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 138 14:667 Schaums and Axiom differ by a constant +ff1:=rootSimp ee1 +--R +--R 2 +--R a +--R (9) - -- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.668~~~~~$\displaystyle @@ -633,10 +2122,13 @@ $$\int{\frac{{\rm sech}^{-1}(x/a)}{x}}= \end{array} \right. $$ + +This is a interesting result since Axiom gives a closed form +solution to the problem but Schaums gives a series solution. <<*>>= )clear all ---S 23 of 32 +--S 139 14:668 SCHAUMS AND AXIOM DIFFER (Axiom has closed form) aa:=integrate(asech(x/a)/x,x) --R --R @@ -660,7 +2152,7 @@ $$ <<*>>= )clear all ---S 24 of 32 +--S 140 aa:=integrate(acsch(x/a),x) --R --R @@ -671,6 +2163,48 @@ aa:=integrate(acsch(x/a),x) --R x --R Type: Union(Expression Integer,...) --E + +--S 141 +bb1:=x*acsch(x/a)+a*asinh(x/a) +--R +--R x x +--R (2) a asinh(-) + x acsch(-) +--R a a +--R Type: Expression Integer +--E + +--S 142 +bb2:=x*acsch(x/a)-a*asinh(x/a) +--R +--R x x +--R (3) - a asinh(-) + x acsch(-) +--R a a +--R Type: Expression Integer +--E + +--S 143 +cc1:=aa-bb1 +--R +--R (4) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a x x +--R - a log(\|x + a - x) + x log(--------------) - a asinh(-) - x acsch(-) +--R x a a +--R Type: Expression Integer +--E + +--S 144 14:669 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a x x +--R - a log(\|x + a - x) + x log(--------------) + a asinh(-) - x acsch(-) +--R x a a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.670~~~~~$\displaystyle @@ -682,7 +2216,7 @@ $$ <<*>>= )clear all ---S 25 of 32 +--S 145 aa:=integrate(x*acsch(x/a),x) --R --R @@ -697,6 +2231,63 @@ aa:=integrate(x*acsch(x/a),x) --R 2\|x + a - 2x --R Type: Union(Expression Integer,...) --E + +--S 146 +bb1:=x^2/2*acsch(x/a)+(a*sqrt(x^2+a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x + a + x acsch(-) +--R a +--R (2) ------------------------ +--R 2 +--R Type: Expression Integer +--E + +--S 147 +bb2:=x^2/2*acsch(x/a)-(a*sqrt(x^2+a^2))/2 +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x + a + x acsch(-) +--R a +--R (3) -------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 148 +cc1:=aa-bb1 +--R +--R +-------+ +--R | 2 2 +--R 2 \|x + a + a 2 x +--R x log(--------------) - x acsch(-) +--R x a +--R (4) ---------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 149 14:670 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R 2 | 2 2 3 \|x + a + a 2 x | 2 2 +--R (x \|x + a - x )log(--------------) + (- x acsch(-) - 2a x)\|x + a +--R x a +--R + +--R 3 x 2 3 +--R x acsch(-) + 2a x + 2a +--R a +--R / +--R +-------+ +--R | 2 2 +--R 2\|x + a - 2x +--R Type: Expression Integer +--E @ \section{\cite{1}:14.671~~~~~$\displaystyle @@ -721,10 +2312,13 @@ $$\int{\frac{{\rm csch}^{-1}(x/a)}{x}}= \end{array} \right. $$ + +Schaums gives 3 different series expansions for this integral +but Axiom has computed a closed form. <<*>>= )clear all ---S 26 of 32 +--S 150 14:671 SCHAUMS AND AXIOM DIFFER (Axiom has closed form) aa:=integrate(acsch(x/a)/x,x) --R --R @@ -737,6 +2331,7 @@ aa:=integrate(acsch(x/a)/x,x) --R 2a --R Type: Union(Expression Integer,...) --E + @ \section{\cite{1}:14.672~~~~~$\displaystyle @@ -748,7 +2343,7 @@ $$ <<*>>= )clear all ---S 27 of 32 +--S 151 14:672 Axiom cannot compute this integral aa:=integrate(x^m*asinh(x/a),x) --R --R @@ -780,7 +2375,7 @@ $$ <<*>>= )clear all ---S 28 of 32 +--S 152 14:673 Axiom cannot compute this integral aa:=integrate(x^m*acosh(x/a),x) --R --R @@ -801,7 +2396,7 @@ $$ <<*>>= )clear all ---S 29 of 32 +--S 153 14:674 Axiom cannot compute this integral aa:=integrate(x^m*atanh(x/a),x) --R --R @@ -822,7 +2417,7 @@ $$ <<*>>= )clear all ---S 30 of 32 +--S 154 14:675 Axiom cannot compute this integral aa:=integrate(x^m*acoth(x/a),x) --R --R @@ -854,7 +2449,7 @@ $$ <<*>>= )clear all ---S 31 of 32 +--S 155 14:676 Axiom cannot compute this integral aa:=integrate(x^m*asech(x/a),x) --R --R @@ -877,7 +2472,7 @@ $$ <<*>>= )clear all ---S 32 of 32 +--S 156 14:677 Axiom cannot compute this integral aa:=integrate(x^m*acsch(x/a),x) --R --R