diff --git a/changelog b/changelog index 29750e1..4b454b8 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20080421 tpd src/input/schaum16.input show Schaums-Axiom equivalence +20080421 tpd src/input/schaum15.input show Schaums-Axiom equivalence +20080421 tpd src/input/schaum14.input show Schaums-Axiom equivalence 20080420 tpd src/input/schaum13.input show Schaums-Axiom equivalence 20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence 20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence diff --git a/src/input/schaum14.input.pamphlet b/src/input/schaum14.input.pamphlet index 532a38a..3452b30 100644 --- a/src/input/schaum14.input.pamphlet +++ b/src/input/schaum14.input.pamphlet @@ -19,7 +19,7 @@ $$ )set message auto off )clear all ---S 1 of 12 +--S 1 aa:=integrate(1/(x^3+a^3),x) --R --R @@ -32,6 +32,41 @@ aa:=integrate(1/(x^3+a^3),x) --R 6a \|3 --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/(6*a^2)*log((x+a)^2/(x^2-a*x+a^2))+1/(a^2*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R 2 2 +-+ +--R x + 2a x + a +-+ (2x - a)\|3 +--R log(--------------) + 2\|3 atan(------------) +--R 2 2 3a +--R x - a x + a +--R (2) --------------------------------------------- +--R 2 +--R 6a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x + 2a x + a +--R - log(x - a x + a ) + 2log(x + a) - log(--------------) +--R 2 2 +--R x - a x + a +--R (3) -------------------------------------------------------- +--R 2 +--R 6a +--R Type: Expression Integer +--E + +--S 4 14:299 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.300~~~~~$\displaystyle @@ -43,7 +78,7 @@ $$ <<*>>= )clear all ---S 2 of 12 +--S 5 aa:=integrate(x/(x^3+a^3),x) --R --R @@ -56,6 +91,39 @@ aa:=integrate(x/(x^3+a^3),x) --R 6a\|3 --R Type: Union(Expression Integer,...) --E + +--S 6 +bb:=1/(6*a)*log((x^2-a*x+a^2)/(x+a)^2)+1/(a*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R 2 2 +-+ +--R x - a x + a +-+ (2x - a)\|3 +--R log(--------------) + 2\|3 atan(------------) +--R 2 2 3a +--R x + 2a x + a +--R (2) --------------------------------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 7 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x - a x + a +--R log(x - a x + a ) - 2log(x + a) - log(--------------) +--R 2 2 +--R x + 2a x + a +--R (3) ------------------------------------------------------ +--R 6a +--R Type: Expression Integer +--E + +--S 8 14:300 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.301~~~~~$\displaystyle @@ -66,7 +134,7 @@ $$ <<*>>= )clear all ---S 3 of 12 +--S 9 aa:=integrate(x^2/(x^3+a^3),x) --R --R @@ -76,6 +144,23 @@ aa:=integrate(x^2/(x^3+a^3),x) --R 3 --R Type: Union(Expression Integer,...) --E + +--S 10 +bb:=1/3*log(x^3+a^3) +--R +--R 3 3 +--R log(x + a ) +--R (2) ------------ +--R 3 +--R Type: Expression Integer +--E + +--S 11 14:301 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.302~~~~~$\displaystyle @@ -86,7 +171,7 @@ $$ <<*>>= )clear all ---S 4 of 12 +--S 12 aa:=integrate(1/(x*(x^3+a^3)),x) --R --R @@ -97,6 +182,41 @@ aa:=integrate(1/(x*(x^3+a^3)),x) --R 3a --R Type: Union(Expression Integer,...) --E + +--S 13 +bb:=1/(3*a^3)*log(x^3/(x^3+a^3)) +--R +--R 3 +--R x +--R log(-------) +--R 3 3 +--R x + a +--R (2) ------------ +--R 3 +--R 3a +--R Type: Expression Integer +--E + +--S 14 +cc:=aa-bb +--R +--R 3 +--R 3 3 x +--R - log(x + a ) + 3log(x) - log(-------) +--R 3 3 +--R x + a +--R (3) --------------------------------------- +--R 3 +--R 3a +--R Type: Expression Integer +--E + +--S 15 14:302 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.303~~~~~$\displaystyle @@ -108,7 +228,7 @@ $$ <<*>>= )clear all ---S 5 of 12 +--S 15 aa:=integrate(1/(x^2*(x^3+a^3)),x) --R --R @@ -121,7 +241,42 @@ aa:=integrate(1/(x^2*(x^3+a^3)),x) --R 4 +-+ --R 6a x\|3 --R Type: Union(Expression Integer,...) +--E + +--S 16 +bb:=-1/(a^3*x)-1/(6*a^4)*log((x^2-a*x+a^2)/(x+a)^2)-1/(a^4*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R 2 2 +-+ +--R x - a x + a +-+ (2x - a)\|3 +--R - x log(--------------) - 2x\|3 atan(------------) - 6a +--R 2 2 3a +--R x + 2a x + a +--R (2) ------------------------------------------------------- +--R 4 +--R 6a x +--R Type: Expression Integer --E + +--S 17 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x - a x + a +--R - log(x - a x + a ) + 2log(x + a) + log(--------------) +--R 2 2 +--R x + 2a x + a +--R (3) -------------------------------------------------------- +--R 4 +--R 6a +--R Type: Expression Integer +--E + +--S 18 14:303 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.304~~~~~$\displaystyle @@ -134,7 +289,7 @@ $$ <<*>>= )clear all ---S 6 of 12 +--S 19 aa:=integrate(1/(x^3+a^3)^2,x) --R --R @@ -151,6 +306,42 @@ aa:=integrate(1/(x^3+a^3)^2,x) --R (9a x + 9a )\|3 --R Type: Union(Expression Integer,...) --E + +--S 20 +bb:=x/(3*a^3*(x^3+a^3))+1/(9*a^5)*log((x+a)^2/(x^2-a*x+a^2))+2/(3*a^5*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R (2) +--R 2 2 +-+ +--R 3 3 x + 2a x + a 3 3 +-+ (2x - a)\|3 2 +--R (x + a )log(--------------) + (2x + 2a )\|3 atan(------------) + 3a x +--R 2 2 3a +--R x - a x + a +--R ----------------------------------------------------------------------- +--R 5 3 8 +--R 9a x + 9a +--R Type: Expression Integer +--E + +--S 21 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x + 2a x + a +--R - log(x - a x + a ) + 2log(x + a) - log(--------------) +--R 2 2 +--R x - a x + a +--R (3) -------------------------------------------------------- +--R 5 +--R 9a +--R Type: Expression Integer +--E + +--S 22 14:304 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.305~~~~~$\displaystyle @@ -163,7 +354,7 @@ $$ <<*>>= )clear all ---S 7 of 12 +--S 23 aa:=integrate(x/(x^3+a^3)^2,x) --R --R @@ -180,6 +371,42 @@ aa:=integrate(x/(x^3+a^3)^2,x) --R (18a x + 18a )\|3 --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=x^2/(3*a^3*(x^3+a^3))+1/(18*a^4)*log((x^2-a*x+a^2)/(x+a)^2)+1/(3*a^4*sqrt(3))*atan((2*x-a)/(a*sqrt(3))) +--R +--R (2) +--R 2 2 +-+ +--R 3 3 x - a x + a 3 3 +-+ (2x - a)\|3 2 +--R (x + a )log(--------------) + (2x + 2a )\|3 atan(------------) + 6a x +--R 2 2 3a +--R x + 2a x + a +--R ------------------------------------------------------------------------ +--R 4 3 7 +--R 18a x + 18a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 x - a x + a +--R log(x - a x + a ) - 2log(x + a) - log(--------------) +--R 2 2 +--R x + 2a x + a +--R (3) ------------------------------------------------------ +--R 4 +--R 18a +--R Type: Expression Integer +--E + +--S 26 14:305 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.306~~~~~$\displaystyle @@ -190,7 +417,7 @@ $$ <<*>>= )clear all ---S 8 of 12 +--S 27 aa:=integrate(x^2/(x^3+a^3)^2,x) --R --R @@ -199,7 +426,24 @@ aa:=integrate(x^2/(x^3+a^3)^2,x) --R 3 3 --R 3x + 3a --R Type: Union(Expression Integer,...) +--E + +--S 28 +bb:=-1/(3*(x^3+a^3)) +--R +--R 1 +--R (2) - --------- +--R 3 3 +--R 3x + 3a +--R Type: Fraction Polynomial Integer --E + +--S 29 14:306 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.307~~~~~$\displaystyle @@ -210,7 +454,7 @@ $$ <<*>>= )clear all ---S 9 of 12 +--S 30 aa:=integrate(1/(x*(x^3+a^3)^2),x) --R --R @@ -221,6 +465,41 @@ aa:=integrate(1/(x*(x^3+a^3)^2),x) --R 3a x + 3a --R Type: Union(Expression Integer,...) --E + +--S 31 +bb:=1/(3*a^3*(x^3+a^3))+1/(3*a^6)*log(x^3/(x^3+a^3)) +--R +--R 3 +--R 3 3 x 3 +--R (x + a )log(-------) + a +--R 3 3 +--R x + a +--R (2) -------------------------- +--R 6 3 9 +--R 3a x + 3a +--R Type: Expression Integer +--E + +--S 32 +cc:=aa-bb +--R +--R 3 +--R 3 3 x +--R - log(x + a ) + 3log(x) - log(-------) +--R 3 3 +--R x + a +--R (3) --------------------------------------- +--R 6 +--R 3a +--R Type: Expression Integer +--E + +--S 33 14:307 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.308~~~~~$\displaystyle @@ -232,7 +511,7 @@ $$ <<*>>= )clear all ---S 10 of 12 +--S 34 aa:=integrate(1/(x^2*(x^3+a^3)^2),x) --R --R @@ -248,7 +527,44 @@ aa:=integrate(1/(x^2*(x^3+a^3)^2),x) --R 7 4 10 +-+ --R (9a x + 9a x)\|3 --R Type: Union(Expression Integer,...) +--E + +--S 35 +t1:=integrate(x/(x^3+a^3),x) +--R +--R +-+ +--R +-+ 2 2 +-+ (2x - a)\|3 +--R \|3 log(x - a x + a ) - 2\|3 log(x + a) + 6atan(------------) +--R 3a +--R (2) -------------------------------------------------------------- +--R +-+ +--R 6a\|3 +--R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=-1/(a^6*x)-x^2/(3*a^6*(x^3+a^3))-4/(3*a^6)*t1 +--R +--R (3) +--R 4 3 +-+ 2 2 4 3 +-+ +--R (- 2x - 2a x)\|3 log(x - a x + a ) + (4x + 4a x)\|3 log(x + a) +--R + +--R +-+ +--R 4 3 (2x - a)\|3 3 4 +-+ +--R (- 12x - 12a x)atan(------------) + (- 12a x - 9a )\|3 +--R 3a +--R / +--R 7 4 10 +-+ +--R (9a x + 9a x)\|3 +--R Type: Expression Integer --E + +--S 37 14:308 Schaums and Axiom agree +cc:=aa-bb +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.309~~~~~$\displaystyle @@ -259,7 +575,7 @@ $$ <<*>>= )clear all ---S 11 of 12 +--S 38 14:309 Axiom cannot compute this integral aa:=integrate(x^m/(x^3+a^3),x) --R --R @@ -269,7 +585,7 @@ aa:=integrate(x^m/(x^3+a^3),x) --R ++ 3 3 --I a + %L --R Type: Union(Expression Integer,...) ---E +--E @ \section{\cite{1}:14.310~~~~~$\displaystyle @@ -280,7 +596,7 @@ $$ <<*>>= )clear all ---S 12 of 12 +--S 39 14:310 Axiom cannot compute this integral aa:=integrate(1/(x^n*(x^3+a^3)),x) --R --R diff --git a/src/input/schaum15.input.pamphlet b/src/input/schaum15.input.pamphlet index 51a6094..0c01fbb 100644 --- a/src/input/schaum15.input.pamphlet +++ b/src/input/schaum15.input.pamphlet @@ -20,7 +20,7 @@ $$ )set message auto off )clear all ---S 1 of 14 +--S 1 aa:=integrate(1/(x^4+a^4),x) --R --R @@ -53,6 +53,202 @@ aa:=integrate(1/(x^4+a^4),x) --R \|2 --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/(4*a^3*sqrt(2))*log((x^2+a*x*sqrt(2)+a^2)/(x^2-a*x*sqrt(2)+a^2))-1/(2*a^3*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2)) +--R +--R +-+ 2 2 +-+ +--R +-+ - a x\|2 - x - a +-+ a x\|2 +--R \|2 log(-------------------) - 2\|2 atan(-------) +--R +-+ 2 2 2 2 +--R a x\|2 - x - a x - a +--R (2) ------------------------------------------------- +--R 3 +--R 8a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R (3) +--R +------+ +------+2 +------+ +--R 3 | 1 8 | 1 4 +-+ | 1 2 +--R 4a |------ log(16a |------ + 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+2 +------+ +--R 3 | 1 8 | 1 4 +-+ | 1 2 +--R - 4a |------ log(16a |------ - 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +--R 4 | 1 +--R 4a |------ +--R +------+ 4| 12 +--R 3 | 1 \|256a +--R 8a |------ atan(--------------------) +--R 4| 12 +------+ +--R \|256a 4 | 1 +-+ +--R 4a |------ - x\|2 +--R 4| 12 +--R \|256a +--R + +--R +------+ +--R 4 | 1 +--R 4a |------ +--R +------+ 4| 12 +-+ 2 2 +--R 3 | 1 \|256a - a x\|2 - x - a +--R - 8a |------ atan(-------------------- - log(-------------------) +--R 4| 12 +------+ +-+ 2 2 +--R \|256a 4 | 1 +-+ a x\|2 - x - a +--R 4a |------ + x\|2 +--R 4| 12 +--R \|256a +--R + +--R +-+ +--R a x\|2 +--R 2atan(-------) +--R 2 2 +--R x - a +--R / +--R 3 +-+ +--R 4a \|2 +--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 (5) +--R +------+ +------+2 +------+ +--R 3 | 1 8 | 1 4 +-+ | 1 2 +--R 4a |------ log(16a |------ + 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+2 +------+ +--R 3 | 1 8 | 1 4 +-+ | 1 2 +--R - 4a |------ log(16a |------ - 4a x\|2 |------ + x ) +--R 4| 12 4| 12 4| 12 +--R \|256a \|256a \|256a +--R + +--R +------+ +--R 4 | 1 +-+ +--R (- 4 + 4%i)a |------ + %i x\|2 +--R +------+ 4| 12 +--R 3 | 1 \|256a +--R 4%i a |------ log(---------------------------------) +--R 4| 12 +------+ +--R \|256a 4 | 1 +-+ +--R (4 + 4%i)a |------ + %i x\|2 +--R 4| 12 +--R \|256a +--R + +--R +------+ +--R 4 | 1 +-+ +--R (- 4 + 4%i)a |------ - %i x\|2 +--R +------+ 4| 12 +--R 3 | 1 \|256a +--R - 4%i a |------ log(---------------------------------) +--R 4| 12 +------+ +--R \|256a 4 | 1 +-+ +--R (4 + 4%i)a |------ - %i x\|2 +--R 4| 12 +--R \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R - a x\|2 + %i x - %i a - a x\|2 - x - a +--R - %i log(-------------------------) - log(-------------------) +--R +-+ 2 2 +-+ 2 2 +--R a x\|2 + %i x - %i a a x\|2 - x - a +--R / +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 6 +ee:=rootSimp dd +--R +--R (6) +--R +-+ +--R +-+ 2 2 x\|2 + (1 + %i)a +--R log(a x\|2 + x + a ) + %i log(-----------------) +--R +-+ +--R x\|2 + (1 - %i)a +--R + +--R +-+ +-+ 2 2 +--R x\|2 + (- 1 - %i)a - a x\|2 + %i x - %i a +--R - %i log(-------------------) - %i log(-------------------------) +--R +-+ +-+ 2 2 +--R x\|2 + (- 1 + %i)a a x\|2 + %i x - %i a +--R + +--R +-+ 2 2 +--R - a x\|2 - x - a +-+ 2 2 +--R - log(-------------------) - log(- a x\|2 + x + a ) +--R +-+ 2 2 +--R a x\|2 - x - a +--R / +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 7 +ff:=expandLog ee +--R +--R (7) +--R +-+ 2 2 +-+ 2 2 +--R %i log(a x\|2 + %i x - %i a ) - %i log(a x\|2 - %i x + %i a ) +--R + +--R +-+ +-+ +--R %i log(x\|2 + (1 + %i)a) - %i log(x\|2 + (1 - %i)a) +--R + +--R +-+ +-+ +--R %i log(x\|2 + (- 1 + %i)a) - %i log(x\|2 + (- 1 - %i)a) +--R + +--R (- 2 - %i)log(- 1) +--R / +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 8 +gg:=complexNormalize ff +--R +--R %i %i +--R %i log(--) - %i log(- --) + (- 2 - %i)log(- 1) +--R 2 2 +--R (8) ---------------------------------------------- +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 9 14:311 Schaums and Axiom differ by a constant +hh:=expandLog gg +--R +--R %i log(%i) - %i log(- %i) + (- 2 - %i)log(- 1) +--R (9) ---------------------------------------------- +--R 3 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.312~~~~~$\displaystyle @@ -63,7 +259,7 @@ $$ <<*>>= )clear all ---S 2 of 14 +--S 10 aa:=integrate(x/(x^4+a^4),x) --R --R @@ -77,6 +273,27 @@ aa:=integrate(x/(x^4+a^4),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 11 +bb:=1/(2*a^2)*atan(x^2/a^2) +--R +--R 2 +--R x +--R atan(--) +--R 2 +--R a +--R (2) -------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 12 14:312 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.313~~~~~$\displaystyle @@ -89,7 +306,7 @@ $$ <<*>>= )clear all ---S 3 of 14 +--S 13 aa:=integrate(x^2/(x^4+a^4),x) --R --R @@ -122,6 +339,213 @@ aa:=integrate(x^2/(x^4+a^4),x) --R \|2 --R Type: Union(Expression Integer,...) --E + +--S 14 +bb:=1/(4*a*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))-1/(2*a*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2)) +--R +--R +-+ 2 2 +-+ +--R +-+ - a x\|2 + x + a +-+ a x\|2 +--R \|2 log(-------------------) - 2\|2 atan(-------) +--R +-+ 2 2 2 2 +--R a x\|2 + x + a x - a +--R (2) ------------------------------------------------- +--R 8a +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R (3) +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R 4a |----- log(- 64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+3 +--R 4 | 1 +--R 64a |----- +--R +-----+ 4| 4 +--R | 1 \|256a +--R 8a |----- atan(---------------------) +--R 4| 4 +-----+3 +--R \|256a 4 | 1 +-+ +--R 64a |----- - x\|2 +--R 4| 4 +--R \|256a +--R + +--R +-----+3 +--R 4 | 1 +--R 64a |----- +--R +-----+ 4| 4 +-+ 2 2 +--R | 1 \|256a - a x\|2 + x + a +--R - 8a |----- atan(--------------------- - log(-------------------) +--R 4| 4 +-----+3 +-+ 2 2 +--R \|256a 4 | 1 +-+ a x\|2 + x + a +--R 64a |----- + x\|2 +--R 4| 4 +--R \|256a +--R + +--R +-+ +--R a x\|2 +--R 2atan(-------) +--R 2 2 +--R x - a +--R / +--R +-+ +--R 4a\|2 +--R Type: Expression Integer +--E + +--S 16 +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 17 +dd:=atanrule cc +--R +--R (5) +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+3 +--R 4 | 1 +-+ +--R (- 64 + 64%i)a |----- + %i x\|2 +--R +-----+ 4| 4 +--R | 1 \|256a +--R 4%i a |----- log(-----------------------------------) +--R 4| 4 +-----+3 +--R \|256a 4 | 1 +-+ +--R (64 + 64%i)a |----- + %i x\|2 +--R 4| 4 +--R \|256a +--R + +--R +-----+3 +--R 4 | 1 +-+ +--R (- 64 + 64%i)a |----- - %i x\|2 +--R +-----+ 4| 4 +--R | 1 \|256a +--R - 4%i a |----- log(-----------------------------------) +--R 4| 4 +-----+3 +--R \|256a 4 | 1 +-+ +--R (64 + 64%i)a |----- - %i x\|2 +--R 4| 4 +--R \|256a +--R + +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R 4a |----- log(- 64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R - a x\|2 + x + a - a x\|2 + %i x - %i a +--R - log(-------------------) - %i log(-------------------------) +--R +-+ 2 2 +-+ 2 2 +--R a x\|2 + x + a a x\|2 + %i x - %i a +--R / +--R +-+ +--R 4a\|2 +--R Type: Expression Complex Integer +--E + +--S 18 +ee:=expandLog dd +--R +--R (6) +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+ +-----+3 +-----+2 +--R | 1 4 +-+ | 1 4 | 1 2 +--R 4a |----- log(64a x\|2 |----- - 16a |----- - x ) +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-----+ +-----+3 +--R | 1 4 | 1 +-+ +--R 4%i a |----- log((64 + 64%i)a |----- + x\|2 ) +--R 4| 4 4| 4 +--R \|256a \|256a +--R + +--R +-----+ +-----+3 +--R | 1 4 | 1 +-+ +--R - 4%i a |----- log((64 + 64%i)a |----- + %i x\|2 ) +--R 4| 4 4| 4 +--R \|256a \|256a +--R + +--R +-----+ +-----+3 +--R | 1 4 | 1 +-+ +--R 4%i a |----- log((64 + 64%i)a |----- - %i x\|2 ) +--R 4| 4 4| 4 +--R \|256a \|256a +--R + +--R +-----+ +-----+3 +-----+ +--R | 1 4 | 1 +-+ | 1 +--R - 4%i a |----- log((64 + 64%i)a |----- - x\|2 + 4a log(- 1) |----- +--R 4| 4 4| 4 4| 4 +--R \|256a \|256a \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R log(a x\|2 + x + a ) + %i log(a x\|2 + %i x - %i a ) +--R + +--R +-+ 2 2 +-+ 2 2 +--R - %i log(a x\|2 - %i x + %i a ) - log(a x\|2 - x - a ) +--R + +--R (- 1 - %i)log(- 1) +--R / +--R +-+ +--R 4a\|2 +--R Type: Expression Complex Integer +--E + +--S 19 +ff:=rootSimp ee +--R +--R (7) +--R +-+ 2 2 +-+ 2 2 +--R %i log(a x\|2 + %i x - %i a ) - %i log(a x\|2 - %i x + %i a ) +--R + +--R +-+ +-+ +--R %i log(x\|2 + (1 + %i)a) - %i log(%i x\|2 + (1 + %i)a) +--R + +--R +-+ +-+ +--R %i log(- %i x\|2 + (1 + %i)a) - %i log(- x\|2 + (1 + %i)a) - %i log(- 1) +--R / +--R +-+ +--R 4a\|2 +--R Type: Expression Complex Integer +--E + +--S 20 14:313 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R %i log(2) - %i log(- 1) - %i log(- 2) +--R (8) ------------------------------------- +--R +-+ +--R 4a\|2 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.314~~~~~$\displaystyle @@ -132,7 +556,7 @@ $$ <<*>>= )clear all ---S 4 of 14 +--S 21 aa:=integrate(x^3/(x^4+a^4),x) --R --R @@ -141,7 +565,24 @@ aa:=integrate(x^3/(x^4+a^4),x) --R (1) ------------ --R 4 --R Type: Union(Expression Integer,...) +--E + +--S 22 +bb:=1/4*log(x^4+a^4) +--R +--R 4 4 +--R log(x + a ) +--R (2) ------------ +--R 4 +--R Type: Expression Integer --E + +--S 23 14:314 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.315~~~~~$\displaystyle @@ -152,7 +593,7 @@ $$ <<*>>= )clear all ---S 5 of 14 +--S 24 aa:=integrate(1/(x*(x^4+a^4)),x) --R --R @@ -163,6 +604,41 @@ aa:=integrate(1/(x*(x^4+a^4)),x) --R 4a --R Type: Union(Expression Integer,...) --E + +--S 25 +bb:=1/(4*a^4)*log(x^4/(x^4+a^4)) +--R +--R 4 +--R x +--R log(-------) +--R 4 4 +--R x + a +--R (2) ------------ +--R 4 +--R 4a +--R Type: Expression Integer +--E + +--S 26 +cc:=aa-bb +--R +--R 4 +--R 4 4 x +--R - log(x + a ) + 4log(x) - log(-------) +--R 4 4 +--R x + a +--R (3) --------------------------------------- +--R 4 +--R 4a +--R Type: Expression Integer +--E + +--S 27 14:315 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.316~~~~~$\displaystyle @@ -175,7 +651,7 @@ $$ <<*>>= )clear all ---S 6 of 14 +--S 28 aa:=integrate(1/(x^2*(x^4+a^4)),x) --R --R @@ -220,6 +696,220 @@ aa:=integrate(1/(x^2*(x^4+a^4)),x) --R a x\|2 --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=-1/(a^4*x)-1/(4*a^5*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))+1/(2*a^5*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2)) +--R +--R +-+ 2 2 +-+ +--R +-+ - a x\|2 + x + a +-+ a x\|2 +--R - x\|2 log(-------------------) + 2x\|2 atan(-------) - 8a +--R +-+ 2 2 2 2 +--R a x\|2 + x + a x - a +--R (2) ---------------------------------------------------------- +--R 5 +--R 8a x +--R Type: Expression Integer +--E + +--S 30 +cc:=aa-bb +--R +--R (3) +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R 4a |------ log(64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R - 4a |------ log(- 64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+3 +--R 16 | 1 +--R 64a |------ +--R +------+ 4| 20 +--R 5 | 1 \|256a +--R - 8a |------ atan(-----------------------) +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R 64a |------ - x\|2 +--R 4| 20 +--R \|256a +--R + +--R +------+3 +--R 16 | 1 +--R 64a |------ +--R +------+ 4| 20 +-+ 2 2 +--R 5 | 1 \|256a - a x\|2 + x + a +--R 8a |------ atan(----------------------- + log(-------------------) +--R 4| 20 +------+3 +-+ 2 2 +--R \|256a 16 | 1 +-+ a x\|2 + x + a +--R 64a |------ + x\|2 +--R 4| 20 +--R \|256a +--R + +--R +-+ +--R a x\|2 +--R - 2atan(-------) +--R 2 2 +--R x - a +--R / +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Integer +--E + +--S 31 +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 32 +dd:=atanrule cc +--R +--R (5) +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R 4a |------ log(64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+3 +--R 16 | 1 +-+ +--R (- 64 + 64%i)a |------ + %i x\|2 +--R +------+ 4| 20 +--R 5 | 1 \|256a +--R - 4%i a |------ log(-------------------------------------) +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R (64 + 64%i)a |------ + %i x\|2 +--R 4| 20 +--R \|256a +--R + +--R +------+3 +--R 16 | 1 +-+ +--R (- 64 + 64%i)a |------ - %i x\|2 +--R +------+ 4| 20 +--R 5 | 1 \|256a +--R 4%i a |------ log(-------------------------------------) +--R 4| 20 +------+3 +--R \|256a 16 | 1 +-+ +--R (64 + 64%i)a |------ - %i x\|2 +--R 4| 20 +--R \|256a +--R + +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R - 4a |------ log(- 64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R - a x\|2 + x + a - a x\|2 + %i x - %i a +--R log(-------------------) + %i log(-------------------------) +--R +-+ 2 2 +-+ 2 2 +--R a x\|2 + x + a a x\|2 + %i x - %i a +--R / +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 33 +ee:=expandLog dd +--R +--R (6) +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R 4a |------ log(64a x\|2 |------ + 16a |------ + x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+3 +------+2 +--R 5 | 1 16 +-+ | 1 12 | 1 2 +--R - 4a |------ log(64a x\|2 |------ - 16a |------ - x ) +--R 4| 20 4| 20 4| 20 +--R \|256a \|256a \|256a +--R + +--R +------+ +------+3 +--R 5 | 1 16 | 1 +-+ +--R - 4%i a |------ log((64 + 64%i)a |------ + x\|2 ) +--R 4| 20 4| 20 +--R \|256a \|256a +--R + +--R +------+ +------+3 +--R 5 | 1 16 | 1 +-+ +--R 4%i a |------ log((64 + 64%i)a |------ + %i x\|2 ) +--R 4| 20 4| 20 +--R \|256a \|256a +--R + +--R +------+ +------+3 +--R 5 | 1 16 | 1 +-+ +--R - 4%i a |------ log((64 + 64%i)a |------ - %i x\|2 ) +--R 4| 20 4| 20 +--R \|256a \|256a +--R + +--R +------+ +------+3 +--R 5 | 1 16 | 1 +-+ +--R 4%i a |------ log((64 + 64%i)a |------ - x\|2 ) +--R 4| 20 4| 20 +--R \|256a \|256a +--R + +--R +------+ +--R 5 | 1 +-+ 2 2 +--R - 4a log(- 1) |------ - log(a x\|2 + x + a ) +--R 4| 20 +--R \|256a +--R + +--R +-+ 2 2 +-+ 2 2 +--R - %i log(a x\|2 + %i x - %i a ) + %i log(a x\|2 - %i x + %i a ) +--R + +--R +-+ 2 2 +--R log(a x\|2 - x - a ) + (1 + %i)log(- 1) +--R / +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 34 +ff:=rootSimp ee +--R +--R (7) +--R +-+ 2 2 +-+ 2 2 +--R - %i log(a x\|2 + %i x - %i a ) + %i log(a x\|2 - %i x + %i a ) +--R + +--R +-+ +-+ +--R - %i log(x\|2 + (1 + %i)a) + %i log(%i x\|2 + (1 + %i)a) +--R + +--R +-+ +-+ +--R - %i log(- %i x\|2 + (1 + %i)a) + %i log(- x\|2 + (1 + %i)a) +--R + +--R %i log(- 1) +--R / +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E + +--S 35 14:316 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R - %i log(2) + %i log(- 1) + %i log(- 2) +--R (8) --------------------------------------- +--R 5 +-+ +--R 4a \|2 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.317~~~~~$\displaystyle @@ -230,7 +920,7 @@ $$ <<*>>= )clear all ---S 7 of 14 +--S 36 aa:=integrate(1/(x^3*(x^4+a^4)),x) --R --R @@ -244,6 +934,27 @@ aa:=integrate(1/(x^3*(x^4+a^4)),x) --R 2a x --R Type: Union(Expression Integer,...) --E + +--S 37 +bb:=-1/(2*a^4*x^2)-1/(2*a^6)*atan(x^2/a^2) +--R +--R 2 +--R 2 x 2 +--R - x atan(--) - a +--R 2 +--R a +--R (2) ----------------- +--R 6 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 38 14:317 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.318~~~~~$\displaystyle @@ -255,7 +966,7 @@ $$ <<*>>= )clear all ---S 8 of 14 +--S 39 aa:=integrate(1/(x^4-a^4),x) --R --R @@ -267,6 +978,37 @@ aa:=integrate(1/(x^4-a^4),x) --R 4a --R Type: Union(Expression Integer,...) --E + +--S 40 +bb:=1/(4*a^3)*log((x-a)/(x+a))-1/(2*a^3)*atan(x/a) +--R +--R x - a x +--R log(-----) - 2atan(-) +--R x + a a +--R (2) --------------------- +--R 3 +--R 4a +--R Type: Expression Integer +--E + +--S 41 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 3 +--R 4a +--R Type: Expression Integer +--E + +--S 42 14:318 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.319~~~~~$\displaystyle @@ -277,7 +1019,7 @@ $$ <<*>>= )clear all ---S 9 of 14 +--S 43 aa:=integrate(x/(x^4-a^4),x) --R --R @@ -288,6 +1030,41 @@ aa:=integrate(x/(x^4-a^4),x) --R 4a --R Type: Union(Expression Integer,...) --E + +--S 44 +bb:=1/(4*a^2)*log((x^2-a^2)/(x^2+a^2)) +--R +--R 2 2 +--R x - a +--R log(-------) +--R 2 2 +--R x + a +--R (2) ------------ +--R 2 +--R 4a +--R Type: Expression Integer +--E + +--S 45 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 2 2 x - a +--R - log(x + a ) + log(x - a ) - log(-------) +--R 2 2 +--R x + a +--R (3) -------------------------------------------- +--R 2 +--R 4a +--R Type: Expression Integer +--E + +--S 46 14:319 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.320~~~~~$\displaystyle @@ -299,7 +1076,7 @@ $$ <<*>>= )clear all ---S 10 of 14 +--S 47 aa:=integrate(x^2/(x^4-a^4),x) --R --R @@ -309,7 +1086,36 @@ aa:=integrate(x^2/(x^4-a^4),x) --R (1) ------------------------------------ --R 4a --R Type: Union(Expression Integer,...) +--E + +--S 48 +bb:=1/(4*a)*log((x-a)/(x+a))+1/(2*a)*atan(x/a) +--R +--R x - a x +--R log(-----) + 2atan(-) +--R x + a a +--R (2) --------------------- +--R 4a +--R Type: Expression Integer --E + +--S 49 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 50 14:320 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.321~~~~~$\displaystyle @@ -320,7 +1126,7 @@ $$ <<*>>= )clear all ---S 11 of 14 +--S 51 aa:=integrate(x^3/(x^4-a^4),x) --R --R @@ -330,6 +1136,23 @@ aa:=integrate(x^3/(x^4-a^4),x) --R 4 --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=1/4*log(x^4-a^4) +--R +--R 4 4 +--R log(x - a ) +--R (2) ------------ +--R 4 +--R Type: Expression Integer +--E + +--S 53 14:321 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.322~~~~~$\displaystyle @@ -340,7 +1163,7 @@ $$ <<*>>= )clear all ---S 12 of 14 +--S 54 aa:=integrate(1/(x*(x^4-a^4)),x) --R --R @@ -350,7 +1173,42 @@ aa:=integrate(1/(x*(x^4-a^4)),x) --R 4 --R 4a --R Type: Union(Expression Integer,...) +--E + +--S 55 +bb:=1/(4*a^4)*log((x^4-a^4)/x^4) +--R +--R 4 4 +--R x - a +--R log(-------) +--R 4 +--R x +--R (2) ------------ +--R 4 +--R 4a +--R Type: Expression Integer --E + +--S 56 +cc:=aa-bb +--R +--R 4 4 +--R 4 4 x - a +--R log(x - a ) - 4log(x) - log(-------) +--R 4 +--R x +--R (3) ------------------------------------- +--R 4 +--R 4a +--R Type: Expression Integer +--E + +--S 57 14:322 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.323~~~~~$\displaystyle @@ -362,7 +1220,7 @@ $$ <<*>>= )clear all ---S 13 of 14 +--S 58 aa:=integrate(1/(x^2*(x^4-a^4)),x) --R --R @@ -374,6 +1232,37 @@ aa:=integrate(1/(x^2*(x^4-a^4)),x) --R 4a x --R Type: Union(Expression Integer,...) --E + +--S 59 +bb:=1/(a^4*x)+1/(4*a^5)*log((x-a)/(x+a))+1/(2*a^5)*atan(x/a) +--R +--R x - a x +--R x log(-----) + 2x atan(-) + 4a +--R x + a a +--R (2) ------------------------------ +--R 5 +--R 4a x +--R Type: Expression Integer +--E + +--S 60 +cc:=aa-bb +--R +--R x - a +--R - log(x + a) + log(x - a) - log(-----) +--R x + a +--R (3) -------------------------------------- +--R 5 +--R 4a +--R Type: Expression Integer +--E + +--S 61 14:323 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.324~~~~~$\displaystyle @@ -384,7 +1273,7 @@ $$ <<*>>= )clear all ---S 14 of 14 +--S 62 aa:=integrate(1/(x^3*(x^4-a^4)),x) --R --R @@ -396,6 +1285,40 @@ aa:=integrate(1/(x^3*(x^4-a^4)),x) --R Type: Union(Expression Integer,...) --E +--S 63 +bb:=1/(2*a^4*x^2)+1/(4*a^6)*log((x^2-a^2)/(x^2+a^2)) +--R +--R 2 2 +--R 2 x - a 2 +--R x log(-------) + 2a +--R 2 2 +--R x + a +--R (2) -------------------- +--R 6 2 +--R 4a x +--R Type: Expression Integer +--E + +--S 64 +cc:=aa-bb +--R +--R 2 2 +--R 2 2 2 2 x - a +--R - log(x + a ) + log(x - a ) - log(-------) +--R 2 2 +--R x + a +--R (3) -------------------------------------------- +--R 6 +--R 4a +--R Type: Expression Integer +--E + +--S 65 14:324 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E )spool )lisp (bye) @ diff --git a/src/input/schaum16.input.pamphlet b/src/input/schaum16.input.pamphlet index f6b849c..b058a54 100644 --- a/src/input/schaum16.input.pamphlet +++ b/src/input/schaum16.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 14 +--S 1 aa:=integrate(1/x*(x^n+a^n),x) --R --R @@ -28,6 +28,79 @@ aa:=integrate(1/x*(x^n+a^n),x) --R n --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/(n*a^n)*log(x^n/(x^n+a^n)) +--R +--R n +--R x +--R log(-------) +--R n n +--R x + a +--R (2) ------------ +--R n +--R n a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R n +--R x n n log(x) n 2 +--R - log(-------) + a %e + n log(x)(a ) +--R n n +--R x + a +--R (3) --------------------------------------------- +--R n +--R n a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R n n n n n log(x) 2n +--R log(x + a ) - log(x ) + a %e + n log(x)a +--R (4) --------------------------------------------------- +--R n +--R n a +--R Type: Expression Integer +--E + +--S 5 +ee:=complexNormalize dd +--R +--R (5) +--R n log(x) n log(a) n log(a) n log(x) +--R log(%e + %e ) + %e %e +--R + +--R n log(a) 2 +--R n log(x)(%e ) - n log(x) +--R / +--R n log(a) +--R n %e +--R Type: Expression Integer +--E + +--S 6 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (6) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 7 14:325 Axiom cannot simplify this expression +ff:=explog ee +--R +--R n n n n 2n +--R log(x + a ) + a x + n log(x)a - n log(x) +--R (7) -------------------------------------------- +--R n +--R n a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.326~~~~~$\displaystyle @@ -38,7 +111,7 @@ $$ <<*>>= )clear all ---S 2 of 14 +--S 8 aa:=integrate(x^(n-1)/(x^n+a^n),x) --R --R @@ -48,6 +121,41 @@ aa:=integrate(x^(n-1)/(x^n+a^n),x) --R n --R Type: Union(Expression Integer,...) --E + +--S 9 +bb:=1/n*log(x^n+a^n) +--R +--R n n +--R log(x + a ) +--R (2) ------------ +--R n +--R Type: Expression Integer +--E + +--S 10 +cc:=aa-bb +--R +--R n log(x) n n n +--R log(%e + a ) - log(x + a ) +--R (3) ----------------------------------- +--R n +--R Type: Expression Integer +--E + +--S 11 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 12 14:326 Schaums and Axiom agree +dd:=explog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.327~~~~~$\displaystyle @@ -59,7 +167,7 @@ $$ <<*>>= )clear all ---S 3 of 14 +--S 13 14:327 Axiom cannot compute this integral aa:=integrate(x^m/(x^n+a^n)^r,x) --R --R @@ -81,7 +189,7 @@ $$ <<*>>= )clear all ---S 4 of 14 +--S 14 14:328 Axiom cannot compute this integral aa:=integrate(1/(x^m*(x^n+a^n)^r),x) --R --R @@ -103,7 +211,7 @@ $$ <<*>>= )clear all ---S 5 of 14 +--S 15 aa:=integrate(1/(x*sqrt(x^n+a^n)),x) --R --R @@ -130,6 +238,114 @@ aa:=integrate(1/(x*sqrt(x^n+a^n)),x) --R n\|- a --R Type: Union(List Expression Integer,...) --E + +--S 16 +bb:=1/(n*sqrt(a^n))*log((sqrt(x^n+a^n)-sqrt(a^n))/(sqrt(x^n+a^n)+sqrt(a^n))) +--R +--R +-------+ +--+ +--R | n n | n +--R \|x + a - \|a +--R log(------------------) +--R +-------+ +--+ +--R | n n | n +--R \|x + a + \|a +--R (2) ----------------------- +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 17 +cc1:=aa.1-bb +--R +--R (3) +--R +---------------+ +--+ +--R n | n log(x) n n log(x) n | n +--R - 2a \|%e + a + (%e + 2a )\|a +--R log(-------------------------------------------------) +--R n log(x) +--R %e +--R + +--R +-------+ +--+ +--R | n n | n +--R \|x + a - \|a +--R - log(------------------) +--R +-------+ +--+ +--R | n n | n +--R \|x + a + \|a +--R / +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 18 +dd1:=expandLog cc1 +--R +--R (4) +--R +---------------+ +--+ +--R n | n log(x) n n log(x) n | n +--R log(2a \|%e + a + (- %e - 2a )\|a ) +--R + +--R +-------+ +--+ +-------+ +--+ +--R | n n | n | n n | n +--R log(\|x + a + \|a ) - log(\|x + a - \|a ) - n log(x) + log(- 1) +--R / +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 19 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (5) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 +ee1:=explog dd1 +--R +--R (6) +--R +-------+ +--+ +-------+ +--+ +--R n | n n n n | n | n n | n +--R log(2a \|x + a + (- x - 2a )\|a ) + log(\|x + a + \|a ) +--R + +--R +-------+ +--+ +--R | n n | n +--R - log(\|x + a - \|a ) - n log(x) + log(- 1) +--R / +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 21 +ff1:=complexNormalize ee1 +--R +--R n log(a) + 4log(- 1) +--R (7) -------------------- +--R +----------+ +--R | n log(a) +--R 2n\|%e +--R Type: Expression Integer +--E + +--S 22 14:329 Schaums and Axiom differ by a constant +gg1:=explog ff1 +--R +--R n log(a) + 4log(- 1) +--R (8) -------------------- +--R +--+ +--R | n +--R 2n\|a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.330~~~~~$\displaystyle @@ -140,7 +356,7 @@ $$ <<*>>= )clear all ---S 6 of 14 +--S 23 aa:=integrate(1/(x*(x^n-a^n)),x) --R --R @@ -151,6 +367,79 @@ aa:=integrate(1/(x*(x^n-a^n)),x) --R n a --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=1/(n*a^n)*log((x^n-a^n)/x^n) +--R +--R n n +--R x - a +--R log(-------) +--R n +--R x +--R (2) ------------ +--R n +--R n a +--R Type: Expression Integer +--E + +--S 25 +cc:=aa-bb +--R +--R n n +--R n log(x) n x - a +--R log(%e - a ) - log(-------) - n log(x) +--R n +--R x +--R (3) ---------------------------------------------- +--R n +--R n a +--R Type: Expression Integer +--E + +--S 26 +dd:=expandLog cc +--R +--R n log(x) n n n n +--R log(%e - a ) + log(x ) - log(x - a ) - n log(x) +--R (4) -------------------------------------------------------- +--R n +--R n a +--R Type: Expression Integer +--E + +--S 27 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (5) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 28 +ee:=explog dd +--R +--R n +--R log(x ) - n log(x) +--R (6) ------------------ +--R n +--R n a +--R Type: Expression Integer +--E + +--S 29 +logpow:=rule(log(a^n) == n*log(a)) +--R +--R n +--R (7) log(a ) == n log(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 30 14:330 Schaums and Axiom agree +ff:=logpow ee +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.331~~~~~$\displaystyle @@ -161,7 +450,7 @@ $$ <<*>>= )clear all ---S 7 of 14 +--S 31 aa:=integrate(x^(n-1)/(x^n-a^n),x) --R --R @@ -171,6 +460,42 @@ aa:=integrate(x^(n-1)/(x^n-a^n),x) --R n --R Type: Union(Expression Integer,...) --E + +--S 32 +bb:=1/n*log(x^n-a^n) +--R +--R n n +--R log(x - a ) +--R (2) ------------ +--R n +--R Type: Expression Integer +--E + +--S 33 +cc:=aa-bb +--R +--R n log(x) n n n +--R log(%e - a ) - log(x - a ) +--R (3) ----------------------------------- +--R n +--R Type: Expression Integer +--E + +--S 34 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 14:331 Schaums and Axiom agree +dd:=explog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.332~~~~~$\displaystyle @@ -182,7 +507,7 @@ $$ <<*>>= )clear all ---S 8 of 14 +--S 36 14:332 Axiom cannot compute this integral aa:=integrate(x^m/(x^n-a^n)^r,x) --R --R @@ -204,7 +529,7 @@ $$ <<*>>= )clear all ---S 9 of 14 +--S 37 14:333 Axiom cannot compute this integral aa:=integrate(1/(x^m*(x^n-a^n)^r),x) --R --R @@ -225,7 +550,7 @@ $$ <<*>>= )clear all ---S 10 of 14 +--S 38 aa:=integrate(1/(x*sqrt(x^n-a^n)),x) --R --R @@ -252,6 +577,62 @@ aa:=integrate(1/(x*sqrt(x^n-a^n)),x) --R n\|a --R Type: Union(List Expression Integer,...) --E + +--S 39 +bb:=2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n)) +--R +--R +--+ +--R | n +--R |a +--R 2acos( |-- ) +--R | n +--R \|x +--R (2) ------------ +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E + +--S 40 +cc1:=aa.1-bb +--R +--R (3) +--R +---------------+ +----+ +--R +--+ n | n log(x) n n log(x) n | n +--R | n 2a \|%e - a + (%e - 2a )\|- a +--R \|a log(-------------------------------------------------) +--R n log(x) +--R %e +--R + +--R +--+ +--R +----+ | n +--R | n |a +--R - 2\|- a acos( |-- ) +--R | n +--R \|x +--R / +--R +----+ +--+ +--R | n | n +--R n\|- a \|a +--R Type: Expression Integer +--E + +--S 41 14:334 Axiom cannot simplify this expression +cc2:=aa.2-bb +--R +--R +--+ +---------------+ +--+ +--R | n | n log(x) n | n +--R \|a \|%e - a |a +--R 2atan(-----------------------) - 2acos( |-- ) +--R n | n +--R a \|x +--R (4) --------------------------------------------- +--R +--+ +--R | n +--R n\|a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.335~~~~~$\displaystyle @@ -269,7 +650,7 @@ $$ <<*>>= )clear all ---S 11 of 14 +--S 42 14:335 Axiom cannot compute this integral aa:=integrate(x^(p-1)/(x^(2*m)+a^(2*m)),x) --R --R @@ -299,7 +680,7 @@ $$ <<*>>= )clear all ---S 12 of 14 +--S 43 14:336 Axiom cannot compute this integral aa:=integrate(x^(p-1)/(x^(2*m)-a^(2*m)),x) --R --R @@ -333,7 +714,7 @@ $$ <<*>>= )clear all ---S 13 of 14 +--S 44 14:337 Axiom cannot compute this integral aa:=integrate(x^(p-1)/(x^(2*m+1)+a^(2*m+1)),x) --R --R @@ -369,7 +750,7 @@ $$ <<*>>= )clear all ---S 14 of 14 +--S 45 14:338 Axiom cannot compute this integral aa:=integrate(x^(p-1)/(x^(2*m+1)-a^(2*m+1)),x) --R --R