diff --git a/changelog b/changelog index 0086e15..7587673 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080430 tpd src/input/schaum13.input post-mortem fixes 20080429 tpd src/input/schaum12.input post-mortem fixes 20080428 tpd src/input/schaum19.input post-mortem fixes 20080428 tpd src/input/schaum23.input post-mortem fixes diff --git a/src/input/schaum13.input.pamphlet b/src/input/schaum13.input.pamphlet index c4a0351..70acff6 100644 --- a/src/input/schaum13.input.pamphlet +++ b/src/input/schaum13.input.pamphlet @@ -31,7 +31,6 @@ $$ --S 1 aa:=integrate(1/sqrt(a*x^2+b*x+c),x) ---R --R --R (1) --R [ @@ -62,14 +61,14 @@ aa:=integrate(1/sqrt(a*x^2+b*x+c),x) --E --S 2 -bb1:=1/sqrt(a)*log(2*sqrt(a)*sqrt(a*x^2+b*x+c)*2*a*x+b) +bb1:=1/sqrt(a)*log(2*sqrt(a)*sqrt(a*x^2+b*x+c)+2*a*x+b) --R ---R +--------------+ ---R +-+ | 2 ---R log(4a x\|a \|a x + b x + c + b) ---R (2) ---------------------------------- ---R +-+ ---R \|a +--R +--------------+ +--R +-+ | 2 +--R log(2\|a \|a x + b x + c + 2a x + b) +--R (2) -------------------------------------- +--R +-+ +--R \|a --R Type: Expression Integer --E @@ -105,9 +104,9 @@ bb3:=1/sqrt(a)*asinh((2*a*x+b)/sqrt(4*a*c-b^2)) cc1:=bb1-aa.1 --R --R (5) ---R +--------------+ ---R +-+ | 2 ---R log(4a x\|a \|a x + b x + c + b) +--R +--------------+ +--R +-+ | 2 +--R log(2\|a \|a x + b x + c + 2a x + b) --R + --R - --R log @@ -131,9 +130,9 @@ cc1:=bb1-aa.1 cc2:=bb1-aa.2 --R --R (6) ---R +--------------+ ---R +---+ +-+ | 2 ---R \|- a log(4a x\|a \|a x + b x + c + b) +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a log(2\|a \|a x + b x + c + 2a x + b) --R + --R +--------------+ --R +---+ | 2 +---+ +-+ @@ -221,7 +220,7 @@ cc5:=bb3-aa.1 --R Type: Expression Integer --E ---S 10 14:280 Axiom cannot simplify these expressions +--S 10 cc6:=bb3-aa.2 --R --R (10) @@ -237,6 +236,42 @@ cc6:=bb3-aa.2 --R \|- a \|a --R Type: Expression Integer --E + +--S 11 +dd1:=simplifyLog cc1 +--R +--R (11) +--R log +--R +--------------+ +--R +-+ +-+ | 2 +--R ((4a x + 2b)\|c + (- 2b x - 4c)\|a )\|a x + b x + c +--R + +--R 2 +-+ +-+ 2 2 +--R (4a x + 4b x + 4c)\|a \|c - 2a b x + (- 4a c - b )x - 2b c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 12 14:280 Schaums and Axiom differ by a constant +ee1:=ratDenom dd1 +--R +--R +-+ +-+ +--R +-+ - 2a\|c + b\|a +--R \|a log(----------------) +--R a +--R (12) ------------------------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.281~~~~~$\displaystyle @@ -499,7 +534,7 @@ cc3:=bb2-aa.1 --R Type: Expression Integer --E ---S 18 14:281 Axiom cannot simplify these expressions +--S 18 cc4:=bb2-aa.2 --R --R +--------------+ @@ -512,6 +547,15 @@ cc4:=bb2-aa.2 --R Type: Expression Integer --E +--S 19 14:281 Schaums and Axiom differ by a constant +dd1:=ratDenom cc4 +--R +--R +-+ +--R \|c +--R (9) - ---- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.282~~~~~$\displaystyle @@ -524,48 +568,85 @@ $$ )clear all --S 19 -aa:=integrate(x^2/(a*x^2+b*x+c),x) ---R +aa:=integrate(x^2/sqrt(a*x^2+b*x+c),x) --R --R (1) --R [ ---R 2 ---R (2a c - b ) +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c --R * --R log ---R +-----------+ ---R 2 2 2 | 2 2 2 ---R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c --R + ---R 3 ---R - 4a b c + b +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a --R / ---R 2 ---R a x + b x + c +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c --R + ---R +-----------+ ---R 2 | 2 ---R (- b log(a x + b x + c) + 2a x)\|- 4a c + b +--R 2 2 3 3 2 2 2 +-+ +--R ((- 16a c - 4a b )x + (- 8a b c + 6b )x + (- 32a c + 24b c)x)\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 +--R 16a b x + (32a c - 8a b )x + (24a b c - 18b )x +--R + +--R 2 2 +--R (32a c - 24b c)x +--R * +--R +-+ +-+ +--R \|a \|c --R / ---R +-----------+ ---R 2 | 2 ---R 2a \|- 4a c + b +--R +--------------+ +--R 2 2 +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a --R , --R ---R +---------+ ---R | 2 ---R 2 (2a x + b)\|4a c - b ---R (- 4a c + 2b )atan(----------------------) ---R 2 ---R 4a c - b +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((- 16a b c + 12b )x - 32a c + 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (16a c - 8a b c - 3b )x + (32a b c - 24b c)x + 32a c - 24b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 2 3 3 2 2 2 +---+ +--R ((- 8a c - 2a b )x + (- 4a b c + 3b )x + (- 16a c + 12b c)x)\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c --R + ---R +---------+ ---R 2 | 2 ---R (- b log(a x + b x + c) + 2a x)\|4a c - b +--R 2 4 2 2 3 3 2 2 2 +--R (8a b x + (16a c - 4a b )x + (12a b c - 9b )x + (16a c - 12b c)x) +--R * +--R +---+ +-+ +--R \|- a \|c --R / ---R +---------+ ---R 2 | 2 ---R 2a \|4a c - b +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (16a b x + 32a c)\|- a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a --R ] --R Type: Union(List Expression Integer,...) --E @@ -652,14 +733,17 @@ bb2:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.2 cc1:=aa.1-bb1 --R --R (5) ---R +-----------+ ---R 2 | 2 ---R (4a c - 3b )\|- 4a c + b +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c --R * --R log --R +--------------+ --R +-+ +-+ | 2 +-+ ---R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c --R + --R 2 +-+ --R (- 2a x - b x - 2c)\|a @@ -668,31 +752,38 @@ cc1:=aa.1-bb1 --R +-+ | 2 --R 2\|c \|a x + b x + c - b x - 2c --R + ---R 2 +-+ ---R (8a c - 4b )\|a +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c --R * --R log ---R +-----------+ ---R 2 2 2 | 2 2 2 ---R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c --R + ---R 3 ---R - 4a b c + b +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a --R / ---R 2 ---R a x + b x + c +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c --R + ---R +-----------+ +--------------+ ---R | 2 +-+ | 2 ---R (- 4a x + 6b)\|- 4a c + b \|a \|a x + b x + c +--R +--------------+ +--R 2 2 +-+ | 2 +--R (- 24b c x - 48b c )\|a \|a x + b x + c --R + ---R +-----------+ ---R 2 | 2 +-+ ---R (- 4b log(a x + b x + c) + 8a x)\|- 4a c + b \|a +--R 3 2 2 2 +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|a \|c --R / ---R +-----------+ ---R 2 | 2 +-+ ---R 8a \|- 4a c + b \|a +--R +--------------+ +--R 2 2 +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a --R Type: Expression Integer --E @@ -700,9 +791,18 @@ cc1:=aa.1-bb1 cc2:=aa.2-bb1 --R --R (6) ---R +---------+ ---R 2 | 2 ---R (4a c - 3b )\|4a c - b +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c +--R + +--R 2 2 +--R 24b c +--R * +--R +---+ +--R \|- a --R * --R log --R +--------------+ @@ -716,24 +816,38 @@ cc2:=aa.2-bb1 --R +-+ | 2 --R 2\|c \|a x + b x + c - b x - 2c --R + ---R +---------+ ---R | 2 ---R 2 +-+ (2a x + b)\|4a c - b ---R (- 16a c + 8b )\|a atan(----------------------) ---R 2 ---R 4a c - b +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 24b )x - 64a c + 48b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (32a c - 16a b c - 6b )x + (64a b c - 48b c)x + 64a c +--R + +--R 2 2 +--R - 48b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x --R + ---R +---------+ +--------------+ ---R +-+ | 2 | 2 ---R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (- 24b c x - 48b c )\|- a \|a \|a x + b x + c --R + ---R +---------+ ---R 2 +-+ | 2 ---R (- 4b log(a x + b x + c) + 8a x)\|a \|4a c - b +--R 3 2 2 2 +---+ +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|- a \|a \|c --R / ---R +---------+ ---R 2 +-+ | 2 ---R 8a \|a \|4a c - b +--R +--------------+ +--R 2 2 +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a --R Type: Expression Integer --E @@ -741,9 +855,18 @@ cc2:=aa.2-bb1 cc3:=aa.2-bb1 --R --R (7) ---R +---------+ ---R 2 | 2 ---R (4a c - 3b )\|4a c - b +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c +--R + +--R 2 2 +--R 24b c +--R * +--R +---+ +--R \|- a --R * --R log --R +--------------+ @@ -757,58 +880,71 @@ cc3:=aa.2-bb1 --R +-+ | 2 --R 2\|c \|a x + b x + c - b x - 2c --R + ---R +---------+ ---R | 2 ---R 2 +-+ (2a x + b)\|4a c - b ---R (- 16a c + 8b )\|a atan(----------------------) ---R 2 ---R 4a c - b +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 24b )x - 64a c + 48b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (32a c - 16a b c - 6b )x + (64a b c - 48b c)x + 64a c +--R + +--R 2 2 +--R - 48b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x --R + ---R +---------+ +--------------+ ---R +-+ | 2 | 2 ---R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (- 24b c x - 48b c )\|- a \|a \|a x + b x + c --R + ---R +---------+ ---R 2 +-+ | 2 ---R (- 4b log(a x + b x + c) + 8a x)\|a \|4a c - b +--R 3 2 2 2 +---+ +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|- a \|a \|c --R / ---R +---------+ ---R 2 +-+ | 2 ---R 8a \|a \|4a c - b +--R +--------------+ +--R 2 2 +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a --R Type: Expression Integer --E ---S 26 14:282 Axiom cannot simplify these expressions +--S 26 cc4:=aa.2-bb2 --R --R (8) ---R +--------------+ ---R +---------+ +---+ | 2 +---+ +-+ ---R 2 | 2 \|- a \|a x + b x + c - \|- a \|c ---R (4a c - 3b )\|4a c - b atan(------------------------------------) ---R a x ---R + ---R +---------+ ---R | 2 ---R 2 +---+ (2a x + b)\|4a c - b ---R (- 8a c + 4b )\|- a atan(----------------------) ---R 2 ---R 4a c - b ---R + ---R +---------+ +--------------+ ---R +---+ | 2 | 2 ---R (- 2a x + 3b)\|- a \|4a c - b \|a x + b x + c ---R + ---R +---------+ ---R 2 +---+ | 2 ---R (- 2b log(a x + b x + c) + 4a x)\|- a \|4a c - b +--R +--------------+ +--R 2 2 | 2 +--R (- 12b c x - 24b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((12a b c + 3b )x + 24b c x + 24b c )\|c --R / ---R +---------+ ---R 2 +---+ | 2 ---R 4a \|- a \|4a c - b +--R +--------------+ +--R 2 2 +-+ | 2 3 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 2 +--R - 32a c --R Type: Expression Integer --E +--S 27 14:282 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R 3b\|c +--R (9) - ------ +--R 2 +--R 4a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.283~~~~~$\displaystyle @@ -832,13 +968,16 @@ $$ )clear all --S 27 -aa:=integrate(1/x*(a*x^2+b*x+c),x) ---R +aa:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R ---R 2 ---R 2c log(x) + a x + 2b x ---R (1) ----------------------- ---R 2 +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (1) -------------------------------------- +--R +-+ +--R \|c --R Type: Union(Expression Integer,...) --E @@ -871,16 +1010,16 @@ bb2:=1/sqrt(-c)*asin((b*x+2*c)/(x*sqrt(b^2-4*a*c))) --E --S 30 -bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(b^2-4*a*c))) ---R ---R b x + 2c ---R asinh(---------------) ---R +-----------+ ---R | 2 ---R x\|- 4a c + b ---R (4) - ---------------------- ---R +-+ ---R \|c +bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(4*a*c-b^2))) +--R +--R b x + 2c +--R asinh(-------------) +--R +---------+ +--R | 2 +--R x\|4a c - b +--R (4) - -------------------- +--R +-+ +--R \|c --R Type: Expression Integer --E @@ -888,42 +1027,96 @@ bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(b^2-4*a*c))) cc1:=aa-bb1 --R --R (5) ---R +--------------+ ---R +-+ | 2 ---R 2\|c \|a x + b x + c + b x + 2c 2 +-+ ---R 2log(---------------------------------) + (2c log(x) + a x + 2b x)\|c ---R x ---R ----------------------------------------------------------------------- ---R +-+ ---R 2\|c +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c + b x + 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R / +--R +-+ +--R \|c --R Type: Expression Integer --E --S 32 cc2:=aa-bb2 --R ---R b x + 2c 2 +---+ ---R - 2asin(---------------) + (2c log(x) + a x + 2b x)\|- c ---R +-----------+ ---R | 2 ---R x\|- 4a c + b ---R (6) ---------------------------------------------------------- ---R +---+ ---R 2\|- c +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +-+ b x + 2c +--R \|- c log(---------------------------------) - \|c asin(---------------) +--R x +-----------+ +--R | 2 +--R x\|- 4a c + b +--R ------------------------------------------------------------------------ +--R +---+ +-+ +--R \|- c \|c --R Type: Expression Integer --E ---S 33 14:283 Axiom cannot simplify these expressions +--S 33 cc3:=aa-bb3 --R ---R b x + 2c 2 +-+ ---R 2asinh(---------------) + (2c log(x) + a x + 2b x)\|c ---R +-----------+ ---R | 2 ---R x\|- 4a c + b ---R (7) ------------------------------------------------------- ---R +-+ ---R 2\|c +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c b x + 2c +--R log(---------------------------------) + asinh(-------------) +--R x +---------+ +--R | 2 +--R x\|4a c - b +--R (7) ------------------------------------------------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 34 +dd1:=expandLog cc1 +--R +--R (8) +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c + b x + 2c) +--R + +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c - b x - 2c) - 2log(x) +--R / +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 35 +ee1:=ratDenom dd1 +--R +--R (9) +--R +--------------+ +--R +-+ +-+ | 2 +--R \|c log(2\|c \|a x + b x + c + b x + 2c) +--R + +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R \|c log(2\|c \|a x + b x + c - b x - 2c) - 2log(x)\|c +--R / +--R c +--R Type: Expression Integer +--E + +--S 36 14:283 Schaums and Axiom differ by a constant +ff1:=complexNormalize ee1 +--R +--R 2 +-+ +--R log(4a c - b )\|c +--R (10) ------------------ +--R c --R Type: Expression Integer --E @ @@ -936,54 +1129,31 @@ $$ <<*>>= )clear all ---S 34 -aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x) ---R +--S 37 +aa:=integrate(1/(x^2*sqrt(a*x^2+b*x+c)),x) --R --R (1) ---R [ ---R 2 ---R (2a c - b )x ---R * ---R log ---R +-----------+ ---R 2 2 2 | 2 2 2 ---R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x ---R + ---R 3 ---R - 4a b c + b ---R / ---R 2 ---R a x + b x + c ---R + ---R +-----------+ ---R 2 | 2 ---R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|- 4a c + b ---R / ---R +-----------+ ---R 2 | 2 ---R 2c x\|- 4a c + b ---R , ---R ---R +---------+ ---R | 2 ---R 2 (2a x + b)\|4a c - b ---R (- 4a c + 2b )x atan(----------------------) ---R 2 ---R 4a c - b ---R + ---R +---------+ ---R 2 | 2 ---R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|4a c - b ---R / ---R +---------+ ---R 2 | 2 ---R 2c x\|4a c - b ---R ] ---R Type: Union(List Expression Integer,...) +--R +--------------+ +--R +-+ | 2 2 2 +--R (- 4b x\|c \|a x + b x + c + 2b x + 4b c x) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (2b x + 8c)\|c \|a x + b x + c + (- 8a c + b )x - 6b c x - 8c +--R / +--R +--------------+ +--R 2 | 2 2 2 +-+ +--R 8c x\|a x + b x + c + (- 4b c x - 8c x)\|c +--R Type: Union(Expression Integer,...) --E ---S 35 +--S 38 t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R --R +--------------+ @@ -997,13 +1167,13 @@ t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R Type: Union(Expression Integer,...) --E ---S 36 -bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*x)*t1 +--S 39 +bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*c)*t1 --R --R +--------------+ --R +-+ | 2 +--------------+ --R 2\|c \|a x + b x + c - b x - 2c +-+ | 2 ---R - b c log(---------------------------------) - 2\|c \|a x + b x + c +--R - b x log(---------------------------------) - 2\|c \|a x + b x + c --R x --R (3) --------------------------------------------------------------------- --R +-+ @@ -1011,64 +1181,104 @@ bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*x)*t1 --R Type: Expression Integer --E ---S 37 -cc1:=aa.1-bb1 +--S 40 +cc:=aa-bb --R --R (4) ---R 2 ---R (2a c - b )x +--R +--------------+ +--R | 2 2 +-+ +--R (4b c\|a x + b x + c + (- 2b x - 4b c)\|c ) --R * ---R log ---R +-----------+ ---R 2 2 2 | 2 2 2 ---R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x ---R + ---R 3 ---R - 4a b c + b ---R / ---R 2 ---R a x + b x + c +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 2 +-+ +--R (- 4b c\|a x + b x + c + (2b x + 4b c)\|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x --R + ---R +-----------+ ---R 2 2 | 2 ---R (b x log(a x + b x + c) - 2b x log(x) - 2bb1 c x - 2c)\|- 4a c + b +--R +--------------+ +--R | 2 2 +-+ +--R - 2b c\|a x + b x + c + (b x + 2b c)\|c --R / ---R +-----------+ ---R 2 | 2 ---R 2c x\|- 4a c + b +--R +--------------+ +--R 2 +-+ | 2 2 3 +--R 8c \|c \|a x + b x + c - 4b c x - 8c --R Type: Expression Integer --E ---S 38 14:284 Axiom cannot simplify these expressions -cc2:=aa.2-bb +--S 41 +dd:=expandLog cc --R --R (5) ---R +--------------+ ---R +---------+ +-+ | 2 ---R 2 | 2 2\|c \|a x + b x + c - b x - 2c ---R b c \|4a c - b log(---------------------------------) ---R x ---R + ---R +---------+ ---R | 2 ---R 2 +-+ (2a x + b)\|4a c - b ---R (- 4a c + 2b )x\|c atan(----------------------) ---R 2 ---R 4a c - b ---R + ---R +---------+ +--------------+ ---R +-+ | 2 | 2 ---R 2c\|c \|4a c - b \|a x + b x + c ---R + ---R +---------+ ---R 2 +-+ | 2 ---R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|c \|4a c - b +--R +--------------+ +--R | 2 2 +-+ +--R (4b c\|a x + b x + c + (- 2b x - 4b c)\|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R | 2 2 +-+ +--R (- 4b c\|a x + b x + c + (2b x + 4b c)\|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +--------------+ +--R | 2 +--R (4b c log(c) + 4b c log(2) - 2b c)\|a x + b x + c +--R + +--R 2 2 2 +-+ +--R ((- 2b x - 4b c)log(c) + (- 2b x - 4b c)log(2) + b x + 2b c)\|c --R / ---R +---------+ ---R 2 +-+ | 2 ---R 2c x\|c \|4a c - b +--R +--------------+ +--R 2 +-+ | 2 2 3 +--R 8c \|c \|a x + b x + c - 4b c x - 8c +--R Type: Expression Integer +--E + +--S 42 +ee:=ratDenom dd +--R +--R (6) +--R +--------------+ +--R +-+ +-+ | 2 +--R 2b\|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R - 2b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (2b log(c) + 2b log(2) - b)\|c +--R / +--R 2 +--R 4c +--R Type: Expression Integer +--E + +--S 43 14:284 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R +-+ +--R (b log(c) + 2b log(2) - b)\|c +--R (7) ------------------------------ +--R 2 +--R 4c --R Type: Expression Integer --E + @ \section{\cite{1}:14.285~~~~~$\displaystyle @@ -1080,7 +1290,7 @@ $$ <<*>>= )clear all ---S 39 +--S 44 aa:=integrate(sqrt(a*x^2+b*x+c),x) --R --R @@ -1165,7 +1375,7 @@ aa:=integrate(sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 40 +--S 45 t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R --R (2) @@ -1196,7 +1406,7 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 41 +--S 46 bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1 --R --R (3) @@ -1224,7 +1434,7 @@ bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1 --R Type: Expression Integer --E ---S 42 +--S 47 bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2 --R --R (4) @@ -1243,7 +1453,7 @@ bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2 --R Type: Expression Integer --E ---S 43 +--S 48 cc1:=aa.1-bb1 --R --R (5) @@ -1263,7 +1473,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 44 +--S 49 cc2:=aa.2-bb1 --R --R (6) @@ -1315,7 +1525,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 45 +--S 50 cc3:=aa.1-bb2 --R --R (7) @@ -1367,7 +1577,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 46 14:285 Axiom cannot simplify these expressions +--S 51 cc4:=aa.2-bb2 --R --R (8) @@ -1386,6 +1596,16 @@ cc4:=aa.2-bb2 --R - 32a c --R Type: Expression Integer --E + +--S 52 14:285 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R b\|c +--R (9) ----- +--R 4a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.286~~~~~$\displaystyle @@ -1403,7 +1623,7 @@ $$ <<*>>= )clear all ---S 47 +--S 53 aa:=integrate(x*sqrt(a*x^2+b*x+c),x) --R --R @@ -1548,7 +1768,7 @@ aa:=integrate(x*sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 48 +--S 54 t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R --R (2) @@ -1579,7 +1799,7 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 49 +--S 55 bb1:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.1 --R --R (3) @@ -1607,7 +1827,7 @@ bb1:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c --R Type: Expression Integer --E ---S 50 +--S 56 bb2:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.2 --R --R (4) @@ -1626,7 +1846,7 @@ bb2:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c --R Type: Expression Integer --E ---S 51 +--S 57 cc1:=aa.1-bb1 --R --R (5) @@ -1720,7 +1940,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 52 +--S 58 cc2:=aa.2-bb1 --R --R (6) @@ -1823,7 +2043,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 53 +--S 59 cc3:=aa.1-bb2 --R --R (7) @@ -1926,7 +2146,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 54 14:286 Axiom cannot simplify these expressions +--S 60 cc4:=aa.2-bb2 --R --R (8) @@ -1957,6 +2177,17 @@ cc4:=aa.2-bb2 --R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x - 768a c --R Type: Expression Integer --E + +--S 61 14:286 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R 2 +-+ +--R (8a c - 3b )\|c +--R (9) ---------------- +--R 2 +--R 24a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.287~~~~~$\displaystyle @@ -1968,7 +2199,7 @@ $$ <<*>>= )clear all ---S 55 +--S 62 aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x) --R --R @@ -2200,7 +2431,7 @@ aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 56 +--S 63 t1:=integrate(sqrt(a*x^2+b*x+c),x) --R --R (2) @@ -2284,7 +2515,7 @@ t1:=integrate(sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 57 +--S 64 bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1 --R --R (3) @@ -2347,7 +2578,7 @@ bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1 --R Type: Expression Integer --E ---S 58 +--S 65 bb2:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.2 --R --R (4) @@ -2404,7 +2635,7 @@ bb2:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.2 --R Type: Expression Integer --E ---S 59 +--S 66 cc1:=aa.1-bb1 --R --R (5) @@ -2600,7 +2831,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 60 +--S 67 cc2:=aa.2-bb1 --R --R (6) @@ -2808,7 +3039,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 61 +--S 68 cc3:=aa.1-bb2 --R --R (7) @@ -3016,7 +3247,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 62 14:287 Axiom cannot simplify these expressions +--S 69 cc4:=aa.2-bb2 --R --R (8) @@ -3083,6 +3314,17 @@ cc4:=aa.2-bb2 --R - 147456a b c x - 49152a c --R Type: Expression Integer --E + +--S 70 14:287 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R 5b c\|c +--R (9) - -------- +--R 2 +--R 24a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.288~~~~~$\displaystyle @@ -3094,7 +3336,7 @@ $$ <<*>>= )clear all ---S 63 +--S 71 aa:=integrate(sqrt(a*x^2+b*x+c)/x,x) --R --R @@ -3178,7 +3420,7 @@ aa:=integrate(sqrt(a*x^2+b*x+c)/x,x) --R Type: Union(List Expression Integer,...) --E ---S 64 +--S 72 t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R --R (2) @@ -3209,7 +3451,7 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 65 +--S 73 t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R --R +--------------+ @@ -3223,7 +3465,7 @@ t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R Type: Union(Expression Integer,...) --E ---S 66 +--S 74 bb1:=sqrt(a*x^2+b*x+c)+b/2*t1.1+c*t2 --R --R (4) @@ -3257,7 +3499,7 @@ bb1:=sqrt(a*x^2+b*x+c)+b/2*t1.1+c*t2 --R Type: Expression Integer --E ---S 67 +--S 75 bb2:=sqrt(a*x^2+b*x+c)+b/2*t1.2+c*t2 --R --R (5) @@ -3282,7 +3524,7 @@ bb2:=sqrt(a*x^2+b*x+c)+b/2*t1.2+c*t2 --R Type: Expression Integer --E ---S 68 +--S 76 cc1:=aa.1-bb1 --R --R (6) @@ -3347,7 +3589,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 69 +--S 77 cc2:=aa.2-bb1 --R --R (7) @@ -3392,7 +3634,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 70 +--S 78 cc3:=aa.1-bb2 --R --R (8) @@ -3444,7 +3686,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 71 14:288 Axiom cannot simplify these expressions +--S 79 cc4:=aa.2-bb2 --R --R (9) @@ -3478,6 +3720,48 @@ cc4:=aa.2-bb2 --R \|- a \|c --R Type: Expression Integer --E + +--S 80 +dd4:=ratDenom cc4 +--R +--R (10) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R - \|c log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 +-+ +--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+ +--R \|c log(--------------------------------------) + \|c +--R 2c x +--R Type: Expression Integer +--E + +--S 81 +ee4:=expandLog dd4 +--R +--R (11) +--R +--------------+ +--R +-+ +-+ | 2 +--R - \|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +-+ +--R \|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) + (- log(c) - log(2) + 1)\|c +--R Type: Expression Integer +--E + +--S 82 14:288 Schaums and Axiom differ by a constant +ff4:=complexNormalize ee4 +--R +--R +-+ +--R (- log(c) - 2log(2) + 2)\|c +--R (12) ---------------------------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.289~~~~~$\displaystyle @@ -3490,7 +3774,7 @@ $$ <<*>>= )clear all ---S 72 +--S 83 aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x) --R --R @@ -3559,7 +3843,7 @@ aa:=integrate(sqrt(a*x^2+b*x+c)/x^2,x) --R Type: Union(List Expression Integer,...) --E ---S 73 +--S 84 t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R --R (2) @@ -3590,7 +3874,7 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 74 +--S 85 t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R --R +--------------+ @@ -3604,7 +3888,7 @@ t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R Type: Union(Expression Integer,...) --E ---S 75 +--S 86 bb1:=-sqrt(a*x^2+b*x+c)/x+a*t1.1+b/2*t2 --R --R (4) @@ -3638,7 +3922,7 @@ bb1:=-sqrt(a*x^2+b*x+c)/x+a*t1.1+b/2*t2 --R Type: Expression Integer --E ---S 76 +--S 87 bb2:=-sqrt(a*x^2+b*x+c)/x+a*t1.2+b/2*t2 --R --R (5) @@ -3663,7 +3947,7 @@ bb2:=-sqrt(a*x^2+b*x+c)/x+a*t1.2+b/2*t2 --R Type: Expression Integer --E ---S 77 +--S 88 cc1:=aa.1-bb1 --R --R (6) @@ -3725,7 +4009,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 78 +--S 89 cc2:=aa.2-bb1 --R --R (7) @@ -3786,7 +4070,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 79 +--S 90 cc3:=aa.1-bb2 --R --R (8) @@ -3842,7 +4126,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 80 14:289 Axiom cannot simplify these expressions +--S 91 cc4:=aa.2-bb2 --R --R (9) @@ -3896,6 +4180,55 @@ cc4:=aa.2-bb2 --R 8c\|- a \|c \|a x + b x + c + (- 4b c x - 8c )\|- a --R Type: Expression Integer --E + +--S 92 +dd4:=ratDenom cc4 +--R +--R (10) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R - 2b\|c log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 +-+ +--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+ +--R 2b\|c log(--------------------------------------) - b\|c +--R 2c x +--R / +--R 4c +--R Type: Expression Integer +--E + +--S 93 +ee4:=expandLog dd4 +--R +--R (11) +--R +--------------+ +--R +-+ +-+ | 2 +--R - 2b\|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R 2b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (- 2b log(c) - 2b log(2) - b)\|c +--R / +--R 4c +--R Type: Expression Integer +--E + +--S 94 14:289 Schaums and Axiom differ by a constant +ff4:=complexNormalize ee4 +--R +--R +-+ +--R (- b log(c) - 2b log(2) - b)\|c +--R (12) -------------------------------- +--R 4c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.290~~~~~$\displaystyle @@ -3907,7 +4240,7 @@ $$ <<*>>= )clear all ---S 81 +--S 95 aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x) --R --R @@ -3921,7 +4254,7 @@ aa:=integrate(1/(a*x^2+b*x+c)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 82 +--S 96 bb:=(2*(2*a*x+b))/((4*a*c-b^2)*sqrt(a*x^2+b*x+c)) --R --R 4a x + 2b @@ -3932,7 +4265,7 @@ bb:=(2*(2*a*x+b))/((4*a*c-b^2)*sqrt(a*x^2+b*x+c)) --R Type: Expression Integer --E ---S 83 14:290 Axiom cannot simplify these expressions +--S 97 cc:=aa-bb --R --R (3) @@ -3945,6 +4278,17 @@ cc:=aa-bb --R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c --R Type: Expression Integer --E + +--S 98 14:290 Schaums and Axiom differ by a constant +dd:=ratDenom cc +--R +--R +-+ +--R 2b\|c +--R (4) ----------- +--R 2 2 +--R 4a c - b c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.291~~~~~$\displaystyle @@ -3955,7 +4299,7 @@ $$ <<*>>= )clear all ---S 84 +--S 99 aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x) --R --R @@ -3968,7 +4312,7 @@ aa:=integrate(x/(a*x^2+b*x+c)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 85 +--S 100 bb:=(2*(b*x+2*c))/((b^2-4*a*c)*sqrt(a*x^2+b*x+c)) --R --R - 2b x - 4c @@ -3979,7 +4323,7 @@ bb:=(2*(b*x+2*c))/((b^2-4*a*c)*sqrt(a*x^2+b*x+c)) --R Type: Expression Integer --E ---S 86 14:291 Axiom cannot simplify these expressions +--S 101 cc:=aa-bb --R --R (3) @@ -3992,6 +4336,17 @@ cc:=aa-bb --R (8a c - 2b c)\|a x + b x + c + ((- 4a b c + b )x - 8a c + 2b c)\|c --R Type: Expression Integer --E + +--S 102 14:291 Schaums and Axiom differ by a constant +dd:=ratDenom cc +--R +--R +-+ +--R 4\|c +--R (4) - --------- +--R 2 +--R 4a c - b +--R Type: Expression Integer +--E @ \section{\cite{1}:14.292~~~~~$\displaystyle @@ -4003,7 +4358,7 @@ $$ <<*>>= )clear all ---S 87 +--S 103 aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x) --R --R @@ -4061,7 +4416,7 @@ aa:=integrate(x^2/(a*x^2+b*x+c)^(3/2),x) --R Type: Union(List Expression Integer,...) --E ---S 88 +--S 104 t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R --R (2) @@ -4092,13 +4447,13 @@ t1:=integrate(1/sqrt(a*x^2+b*x+c),x) --R Type: Union(List Expression Integer,...) --E ---S 89 -bb1:=((2*b^2-4*a*c)*x+2*b*c)/(4*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.1 +--S 105 +bb1:=((2*b^2-4*a*c)*x+2*b*c)/(a*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.1 --R --R (3) ---R +--------------+ ---R 2 | 2 ---R (8a c - 2b )\|a x + b x + c +--R +--------------+ +--R 2 | 2 +--R (4a c - b )\|a x + b x + c --R * --R log --R +--------------+ @@ -4112,94 +4467,58 @@ bb1:=((2*b^2-4*a*c)*x+2*b*c)/(4*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.1 --R +-+ | 2 --R 2\|c \|a x + b x + c - b x - 2c --R + ---R 2 2 +-+ ---R ((- 2a c + a b )x + a b c)\|a +--R 2 +-+ +--R ((- 4a c + 2b )x + 2b c)\|a --R / ---R +--------------+ ---R 2 2 +-+ | 2 ---R (8a c - 2a b )\|a \|a x + b x + c +--R +--------------+ +--R 2 2 +-+ | 2 +--R (4a c - a b )\|a \|a x + b x + c --R Type: Expression Integer --E ---S 90 -bb2:=((2*b^2-4*a*c)*x+2*b*c)/(4*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.2 +--S 106 +bb2:=((2*b^2-4*a*c)*x+2*b*c)/(a*(4*a*c-b^2)*sqrt(a*x^2+b*x+c))+1/a*t1.2 --R --R (4) ---R +--------------+ ---R +--------------+ +---+ | 2 +---+ +-+ ---R 2 | 2 \|- a \|a x + b x + c - \|- a \|c ---R (16a c - 4b )\|a x + b x + c atan(------------------------------------) ---R a x ---R + ---R 2 2 +---+ ---R ((- 2a c + a b )x + a b c)\|- a +--R +--------------+ +--R +--------------+ +---+ | 2 +---+ +-+ +--R 2 | 2 \|- a \|a x + b x + c - \|- a \|c +--R (8a c - 2b )\|a x + b x + c atan(------------------------------------) +--R a x +--R + +--R 2 +---+ +--R ((- 4a c + 2b )x + 2b c)\|- a --R / ---R +--------------+ ---R 2 2 +---+ | 2 ---R (8a c - 2a b )\|- a \|a x + b x + c +--R +--------------+ +--R 2 2 +---+ | 2 +--R (4a c - a b )\|- a \|a x + b x + c --R Type: Expression Integer --E ---S 91 +--S 107 cc1:=aa.1-bb1 --R --R (5) ---R 2 3 2 2 2 2 ---R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b c)x ---R + ---R 2 ---R 2a b c ---R * ---R +--------------+ ---R +-+ | 2 ---R \|c \|a x + b x + c ---R + ---R 3 2 2 2 2 3 ---R ((4a - 16a )c + (- 2a + 4a)b c)x ---R + ---R 2 2 3 2 2 3 2 2 ---R ((2a - 16a)b c + (- 2a + 4)b c)x + ((4a - 16a)c + (- 4a + 4)b c )x ---R + ---R 3 ---R - 2a b c ---R / ---R 3 2 2 2 2 2 2 3 2 3 2 2 ---R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c ) ---R * ---R +--------------+ ---R | 2 ---R \|a x + b x + c ---R + ---R 3 2 3 3 3 2 2 2 4 2 ---R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x ---R + ---R 2 2 3 2 3 2 2 ---R (- 24a b c + 6a b c)x - 16a c + 4a b c ---R * ---R +-+ ---R \|c +--R +--------------+ +--R +-+ | 2 2 2 +--R 4b c\|c \|a x + b x + c - 2b c x - 4b c +--R ----------------------------------------------------------------------------- +--R +--------------+ +--R 2 2 2 | 2 2 3 2 2 2 +-+ +--R (8a c - 2a b c)\|a x + b x + c + ((- 4a b c + a b )x - 8a c + 2a b c)\|c --R Type: Expression Integer --E ---S 92 +--S 108 cc2:=aa.2-bb1 --R --R (6) ---R 2 2 2 2 2 3 3 2 2 ---R ((- 16a c + 4a b c)x + (- 16a b c + 4b c)x - 16a c + 4b c ) ---R * ---R +--------------+ ---R +---+ | 2 ---R \|- a \|a x + b x + c +--R +--------------+ +--R 2 2 +---+ | 2 +--R (- 8a c + 2b c)\|- a \|a x + b x + c --R + ---R 2 3 3 2 2 2 4 2 ---R (8a b c - 2a b )x + (16a c + 4a b c - 2b )x ---R + ---R 2 3 3 2 2 ---R (24a b c - 6b c)x + 16a c - 4b c ---R * ---R +---+ +-+ ---R \|- a \|c +--R 3 2 2 +---+ +-+ +--R ((4a b c - b )x + 8a c - 2b c)\|- a \|c --R * --R log --R +--------------+ @@ -4213,21 +4532,12 @@ cc2:=aa.2-bb1 --R +-+ | 2 --R 2\|c \|a x + b x + c - b x - 2c --R + ---R 2 2 2 2 2 3 3 2 2 +-+ ---R ((32a c - 8a b c)x + (32a b c - 8b c)x + 32a c - 8b c )\|a ---R * ---R +--------------+ ---R | 2 ---R \|a x + b x + c +--R +--------------+ +--R 2 2 +-+ | 2 +--R (16a c - 4b c)\|a \|a x + b x + c --R + ---R 2 3 3 2 2 2 4 2 ---R (- 16a b c + 4a b )x + (- 32a c - 8a b c + 4b )x ---R + ---R 2 3 3 2 2 ---R (- 48a b c + 12b c)x - 32a c + 8b c ---R * ---R +-+ +-+ ---R \|a \|c +--R 3 2 2 +-+ +-+ +--R ((- 8a b c + 2b )x - 16a c + 4b c)\|a \|c --R * --R +--------------+ --R +---+ | 2 +---+ +-+ @@ -4235,65 +4545,29 @@ cc2:=aa.2-bb1 --R atan(------------------------------------) --R a x --R + ---R 2 3 2 2 2 2 ---R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b c)x ---R + ---R 2 ---R 2a b c ---R * ---R +--------------+ ---R +---+ +-+ +-+ | 2 ---R \|- a \|a \|c \|a x + b x + c ---R + ---R 3 2 2 2 2 3 ---R ((4a - 16a )c + (- 2a + 4a)b c)x ---R + ---R 2 2 3 2 ---R ((2a - 16a)b c + (- 2a + 4)b c)x ---R + ---R 2 3 2 2 3 ---R ((4a - 16a)c + (- 4a + 4)b c )x - 2a b c ---R * ---R +---+ +-+ ---R \|- a \|a +--R +--------------+ +--R +---+ +-+ +-+ | 2 2 2 +---+ +-+ +--R 4b c\|- a \|a \|c \|a x + b x + c + (- 2b c x - 4b c )\|- a \|a --R / ---R 3 2 2 2 2 2 2 3 2 3 2 2 +---+ ---R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c )\|- a ---R * ---R +--------------+ ---R +-+ | 2 ---R \|a \|a x + b x + c +--R +--------------+ +--R 2 2 2 +---+ +-+ | 2 +--R (8a c - 2a b c)\|- a \|a \|a x + b x + c --R + ---R 3 2 3 3 3 2 2 2 4 2 ---R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x ---R + ---R 2 2 3 2 3 2 2 ---R (- 24a b c + 6a b c)x - 16a c + 4a b c ---R * ---R +---+ +-+ +-+ ---R \|- a \|a \|c +--R 2 3 2 2 2 +---+ +-+ +-+ +--R ((- 4a b c + a b )x - 8a c + 2a b c)\|- a \|a \|c --R Type: Expression Integer --E ---S 93 +--S 109 cc3:=aa.1-bb2 --R --R (7) ---R 2 2 2 2 2 3 3 2 2 +---+ ---R ((16a c - 4a b c)x + (16a b c - 4b c)x + 16a c - 4b c )\|- a ---R * ---R +--------------+ ---R | 2 ---R \|a x + b x + c +--R +--------------+ +--R 2 2 +---+ | 2 +--R (8a c - 2b c)\|- a \|a x + b x + c --R + ---R 2 3 3 2 2 2 4 2 ---R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2b )x ---R + ---R 2 3 3 2 2 ---R (- 24a b c + 6b c)x - 16a c + 4b c ---R * ---R +---+ +-+ ---R \|- a \|c +--R 3 2 2 +---+ +-+ +--R ((- 4a b c + b )x - 8a c + 2b c)\|- a \|c --R * --R log --R +--------------+ @@ -4307,21 +4581,12 @@ cc3:=aa.1-bb2 --R +-+ | 2 --R 2\|c \|a x + b x + c - b x - 2c --R + ---R 2 2 2 2 2 3 3 2 2 +-+ ---R ((- 32a c + 8a b c)x + (- 32a b c + 8b c)x - 32a c + 8b c )\|a ---R * ---R +--------------+ ---R | 2 ---R \|a x + b x + c +--R +--------------+ +--R 2 2 +-+ | 2 +--R (- 16a c + 4b c)\|a \|a x + b x + c --R + ---R 2 3 3 2 2 2 4 2 ---R (16a b c - 4a b )x + (32a c + 8a b c - 4b )x ---R + ---R 2 3 3 2 2 ---R (48a b c - 12b c)x + 32a c - 8b c ---R * ---R +-+ +-+ ---R \|a \|c +--R 3 2 2 +-+ +-+ +--R ((8a b c - 2b )x + 16a c - 4b c)\|a \|c --R * --R +--------------+ --R +---+ | 2 +---+ +-+ @@ -4329,87 +4594,43 @@ cc3:=aa.1-bb2 --R atan(------------------------------------) --R a x --R + ---R 2 3 2 2 2 2 ---R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b c)x ---R + ---R 2 ---R 2a b c ---R * ---R +--------------+ ---R +---+ +-+ +-+ | 2 ---R \|- a \|a \|c \|a x + b x + c ---R + ---R 3 2 2 2 2 3 ---R ((4a - 16a )c + (- 2a + 4a)b c)x ---R + ---R 2 2 3 2 ---R ((2a - 16a)b c + (- 2a + 4)b c)x ---R + ---R 2 3 2 2 3 ---R ((4a - 16a)c + (- 4a + 4)b c )x - 2a b c ---R * ---R +---+ +-+ ---R \|- a \|a +--R +--------------+ +--R +---+ +-+ +-+ | 2 2 2 +---+ +-+ +--R 4b c\|- a \|a \|c \|a x + b x + c + (- 2b c x - 4b c )\|- a \|a --R / ---R 3 2 2 2 2 2 2 3 2 3 2 2 +---+ ---R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c )\|- a ---R * ---R +--------------+ ---R +-+ | 2 ---R \|a \|a x + b x + c +--R +--------------+ +--R 2 2 2 +---+ +-+ | 2 +--R (8a c - 2a b c)\|- a \|a \|a x + b x + c --R + ---R 3 2 3 3 3 2 2 2 4 2 ---R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x ---R + ---R 2 2 3 2 3 2 2 ---R (- 24a b c + 6a b c)x - 16a c + 4a b c ---R * ---R +---+ +-+ +-+ ---R \|- a \|a \|c +--R 2 3 2 2 2 +---+ +-+ +-+ +--R ((- 4a b c + a b )x - 8a c + 2a b c)\|- a \|a \|c --R Type: Expression Integer --E ---S 94 14:292 Axiom cannot simplify these expressions +--S 110 cc4:=aa.2-bb2 --R --R (8) ---R 2 3 2 2 2 2 ---R ((- 2a + 16a)b c + (a - 4)b )x + ((- 4a + 16a)c + (3a - 4)b c)x ---R + ---R 2 ---R 2a b c ---R * ---R +--------------+ ---R +-+ | 2 ---R \|c \|a x + b x + c ---R + ---R 3 2 2 2 2 3 ---R ((4a - 16a )c + (- 2a + 4a)b c)x ---R + ---R 2 2 3 2 2 3 2 2 ---R ((2a - 16a)b c + (- 2a + 4)b c)x + ((4a - 16a)c + (- 4a + 4)b c )x ---R + ---R 3 ---R - 2a b c ---R / ---R 3 2 2 2 2 2 2 3 2 3 2 2 ---R ((16a c - 4a b c)x + (16a b c - 4a b c)x + 16a c - 4a b c ) ---R * ---R +--------------+ ---R | 2 ---R \|a x + b x + c ---R + ---R 3 2 3 3 3 2 2 2 4 2 ---R (- 8a b c + 2a b )x + (- 16a c - 4a b c + 2a b )x ---R + ---R 2 2 3 2 3 2 2 ---R (- 24a b c + 6a b c)x - 16a c + 4a b c ---R * ---R +-+ ---R \|c +--R +--------------+ +--R +-+ | 2 2 2 +--R 4b c\|c \|a x + b x + c - 2b c x - 4b c +--R ----------------------------------------------------------------------------- +--R +--------------+ +--R 2 2 2 | 2 2 3 2 2 2 +-+ +--R (8a c - 2a b c)\|a x + b x + c + ((- 4a b c + a b )x - 8a c + 2a b c)\|c --R Type: Expression Integer --E +--S 111 14:292 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R 2b\|c +--R (9) ----------- +--R 2 2 +--R 4a c - a b +--R Type: Expression Integer +--E @ \section{\cite{1}:14.293~~~~~$\displaystyle @@ -4422,7 +4643,7 @@ $$ <<*>>= )clear all ---S 95 +--S 112 aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x) --R --R @@ -4448,7 +4669,7 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 96 +--S 113 t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R --R +--------------+ @@ -4462,7 +4683,7 @@ t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R Type: Union(Expression Integer,...) --E ---S 97 +--S 114 t2:=integrate(1/(a*x^2+b*x+c)^(3/2),x) --R --R +--------------+ @@ -4475,7 +4696,7 @@ t2:=integrate(1/(a*x^2+b*x+c)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 98 +--S 115 bb:=1/(c*sqrt(a*x^2+b*x+c))+1/c*t1-b/(2*c)*t2 --R --R (4) @@ -4505,7 +4726,7 @@ bb:=1/(c*sqrt(a*x^2+b*x+c))+1/c*t1-b/(2*c)*t2 --R Type: Expression Integer --E ---S 99 14:293 Axiom cannot simplify these expressions +--S 116 cc:=aa-bb --R --R (5) @@ -4551,6 +4772,58 @@ cc:=aa-bb --R (- 8a c - 4b c )x - 12b c x - 8c --R Type: Expression Integer --E + +--S 117 +dd:=ratDenom cc +--R +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R - \|c log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 +-+ +--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+ +--R \|c log(--------------------------------------) + \|c +--R 2c x +--R / +--R 2 +--R c +--R Type: Expression Integer +--E + +--S 118 +ee:=expandLog dd +--R +--R (7) +--R +--------------+ +--R +-+ +-+ | 2 +--R - \|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R \|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (- log(c) - log(2) + 1)\|c +--R / +--R 2 +--R c +--R Type: Expression Integer +--E + +--S 119 14:293 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R +-+ +--R (- log(c) - 2log(2) + 2)\|c +--R (8) ---------------------------- +--R 2 +--R 2c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.294~~~~~$\displaystyle @@ -4567,7 +4840,7 @@ $$ <<*>>= )clear all ---S 100 +--S 120 aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x) --R --R @@ -4608,7 +4881,7 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 101 +--S 121 t1:=integrate(1/(a*x^2+b*x+c)^(3/2),x) --R --R +--------------+ @@ -4621,7 +4894,7 @@ t1:=integrate(1/(a*x^2+b*x+c)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 102 +--S 122 t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R --R +--------------+ @@ -4635,8 +4908,8 @@ t2:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) --R Type: Union(Expression Integer,...) --E ---S 103 -bb:=-(a*x^2+b*x+c)/(c^2*x*sqrt(a*x^2+b*x+c))+(b^2-2*a*c)/(2*c^2)*t1-(3*b)/(2*c^2)*t2 +--S 123 +bb:=-(a*x^2+2*b*x+c)/(c^2*x*sqrt(a*x^2+b*x+c))+(b^2-2*a*c)/(2*c^2)*t1-(3*b)/(2*c^2)*t2 --R --R (4) --R +--------------+ @@ -4652,15 +4925,15 @@ bb:=-(a*x^2+b*x+c)/(c^2*x*sqrt(a*x^2+b*x+c))+(b^2-2*a*c)/(2*c^2)*t1-(3*b)/(2*c^2 --R log(---------------------------------) --R x --R + ---R +--------------+ ---R 3 2 2 2 3 | 2 ---R (2a b c x + 8a c x + 6b c x + 4c )\|a x + b x + c +--R +--------------+ +--R 3 2 2 2 2 3 | 2 +--R (2a b c x + (8a c + 2b c)x + 10b c x + 4c )\|a x + b x + c --R + --R 2 2 4 3 3 2 2 2 ---R (- 8a c + 2a b )x + (- 12a b c + 2b )x + (- 12a c - 2b c)x +--R (- 8a c + 2a b )x + (- 16a b c + 2b )x + (- 12a c - 6b c)x --R + ---R 2 3 ---R - 8b c x - 4c +--R 2 3 +--R - 12b c x - 4c --R * --R +-+ --R \|c @@ -4674,7 +4947,7 @@ bb:=-(a*x^2+b*x+c)/(c^2*x*sqrt(a*x^2+b*x+c))+(b^2-2*a*c)/(2*c^2)*t1-(3*b)/(2*c^2 --R Type: Expression Integer --E ---S 104 14:284 Axiom cannot simplify these expressions +--S 124 cc:=aa-bb --R --R (5) @@ -4720,18 +4993,18 @@ cc:=aa-bb --R +-+ --R 2x\|c --R + ---R 2 4 3 2 3 2 2 2 3 +-+ ---R ((- 60a b c - 5b )x + (- 88a b c - 66b c)x - 112b c x - 32b c )\|c +--R 2 4 3 2 3 2 2 2 3 +--R ((- 60a b c - 5b )x + (- 120a b c - 90b c)x - 240b c x - 160b c ) --R * ---R +--------------+ ---R | 2 ---R \|a x + b x + c +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c --R + --R 2 2 3 4 2 2 4 3 3 3 2 2 ---R (40a b c + 30a b c)x + (152a b c + 26b c)x + (104a b c + 118b c )x +--R (40a b c + 30a b c)x + (200a b c + 30b c)x + (200a b c + 190b c )x --R + ---R 2 3 4 ---R 128b c x + 32b c +--R 2 3 4 +--R 320b c x + 160b c --R / --R 4 3 3 3 5 2 4 2 5 6 --R ((48a b c + 4b c )x + (96a c + 72b c )x + 192b c x + 128c ) @@ -4750,6 +5023,58 @@ cc:=aa-bb --R \|c --R Type: Expression Integer --E + +--S 125 +dd:=ratDenom cc +--R +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +-+ 2\|c \|a x + b x + c - b x - 2c +--R 6b\|c log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 +-+ +--R +-+ 2c\|a x + b x + c + (- b x - 2c)\|c +-+ +--R - 6b\|c log(--------------------------------------) - 5b\|c +--R 2c x +--R / +--R 3 +--R 4c +--R Type: Expression Integer +--E + +--S 126 +ee:=expandLog dd +--R +--R (7) +--R +--------------+ +--R +-+ +-+ | 2 +--R 6b\|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R - 6b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (6b log(c) + 6b log(2) - 5b)\|c +--R / +--R 3 +--R 4c +--R Type: Expression Integer +--E + +--S 127 14:284 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R +-+ +--R (3b log(c) + 6b log(2) - 5b)\|c +--R (8) -------------------------------- +--R 3 +--R 4c +--R Type: Expression Integer +--E @ \section{\cite{1}:14.295~~~~~$\displaystyle @@ -4765,7 +5090,7 @@ $$ <<*>>= )clear all ---S 105 14:295 Axiom cannot compute this integral +--S 128 14:295 Axiom cannot compute this integral aa:=integrate((a*x^2+b*x+c)^(n+1/2),x) --R --R @@ -4787,7 +5112,7 @@ $$ <<*>>= )clear all ---S 106 14:296 Axiom cannot compute this integral +--S 129 14:296 Axiom cannot compute this integral aa:=integrate(x*(a*x^2+b*x+c)^(n+1/2),x) --R --R @@ -4813,7 +5138,7 @@ $$ <<*>>= )clear all ---S 107 14:297 Axiom cannot compute this integral +--S 130 14:297 Axiom cannot compute this integral aa:=integrate(1/(a*x^2+b*x+c)^(n+1/2),x) --R --R @@ -4844,7 +5169,7 @@ $$ <<*>>= )clear all ---S 108 14:298 Axiom cannot compute this integral +--S 131 14:298 Axiom cannot compute this integral aa:=integrate(1/(x*(a*x^2+b*x+c)^(n+1/2)),x) --R --R