diff --git a/changelog b/changelog index c10550c..d61af40 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence 20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence 20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence 20080418 tpd src/input/schaum9.input fix Schaums-Axiom equivalence diff --git a/src/input/schaum12.input.pamphlet b/src/input/schaum12.input.pamphlet index 644ebed..1c0d0b2 100644 --- a/src/input/schaum12.input.pamphlet +++ b/src/input/schaum12.input.pamphlet @@ -24,7 +24,7 @@ $$ )set message auto off )clear all ---S 1 of 15 +--S 1 aa:=integrate(1/sqrt(a*x^2+b*x+c),x) --R --R @@ -55,6 +55,152 @@ aa:=integrate(1/sqrt(a*x^2+b*x+c),x) --R \|- a --R Type: Union(List Expression Integer,...) --E + +--S 2 +bb1:=2/sqrt(4*a*c-b^2)*atan((2*a*x+b)/sqrt(4*a*c-b^2)) +--R +--R +--R 2a x + b +--R 2atan(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R (2) ------------------- +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/sqrt(b^2-4*a*c)*log((2*a*x+b-sqrt(b^2-4*a*c))/(2*a*x+b+sqrt(b^2-4*a*c))) +--R +--R +--R +-----------+ +--R | 2 +--R - \|- 4a c + b + 2a x + b +--R log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R (3) -------------------------------- +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa.1-bb1 +--R +--R (4) +--R +---------+ +--R | 2 +--R \|4a c - b +--R * +--R log +--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 +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +-+ 2a x + b +--R - 2\|a atan(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R +-+ | 2 +--R \|a \|4a c - b +--R Type: Expression Integer +--E + +--S 5 +cc2:=aa.1-bb2 +--R +--R (5) +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R * +--R log +--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 +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +-----------+ +--R | 2 +--R +-+ - \|- 4a c + b + 2a x + b +--R - \|a log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R / +--R +-----------+ +--R | 2 +-+ +--R \|- 4a c + b \|a +--R Type: Expression Integer +--E + +--S 6 +cc3:=aa.2-bb1 +--R +--R (6) +--R +--------------+ +--R +---------+ +---+ | 2 +---+ +-+ +--R | 2 \|- a \|a x + b x + c - \|- a \|c +--R 2\|4a c - b atan(------------------------------------) +--R a x +--R + +--R +---+ 2a x + b +--R - 2\|- a atan(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R +---+ | 2 +--R \|- a \|4a c - b +--R Type: Expression Integer +--E + +--S 7 14:265 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (7) +--R +-----------+ +--R | 2 +--R +---+ - \|- 4a c + b + 2a x + b +--R - \|- a log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R + +--R +--------------+ +--R +-----------+ +---+ | 2 +---+ +-+ +--R | 2 \|- a \|a x + b x + c - \|- a \|c +--R 2\|- 4a c + b atan(------------------------------------) +--R a x +--R / +--R +-----------+ +--R | 2 +---+ +--R \|- 4a c + b \|- a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.266~~~~~$\displaystyle @@ -65,7 +211,7 @@ $$ <<*>>= )clear all ---S 2 of 15 +--S 8 aa:=integrate(x/sqrt(a*x^2+b*x+c),x) --R --R @@ -116,6 +262,316 @@ aa:=integrate(x/sqrt(a*x^2+b*x+c),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 9 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--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 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 10 +bb1:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.1 +--R +--R +--R (3) +--R - +--R b +--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 log(a x + b x + c)\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2a\|- 4a c + b +--R Type: Expression Integer +--E + +--S 11 +bb2:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.2 +--R +--R +--R +---------+ +--R | 2 +---------+ +--R (2a x + b)\|4a c - b 2 | 2 +--R - 2b atan(----------------------) + log(a x + b x + c)\|4a c - b +--R 2 +--R 4a c - b +--R (4) ------------------------------------------------------------------- +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R Type: Expression Integer +--E + +--S 12 +cc1:=aa.1-bb1 +--R +--R (5) +--R +-----------+ +--------------+ +-----------+ +--R | 2 +-+ | 2 2 | 2 +--R (2b\|- 4a c + b \|c \|a x + b x + c + (- b x - 2b c)\|- 4a c + b ) +--R * +--R log +--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 +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R (2b\|a \|c \|a x + b x + c + (- b x - 2b c)\|a ) +--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 +-+ +-+ | 2 +-+ +--R (- 2log(a x + b x + c)\|- 4a c + b \|a \|c - 2b x\|- 4a c + b \|a ) +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +-+ +-+ +--R (4a x + 2b x)\|- 4a c + b \|a \|c +--R + +--R +-----------+ +--R 2 | 2 +-+ +--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|a +--R / +--R +-----------+ +--------------+ +--R | 2 +-+ +-+ | 2 +--R 4a\|- 4a c + b \|a \|c \|a x + b x + c +--R + +--R +-----------+ +--R | 2 +-+ +--R (- 2a b x - 4a c)\|- 4a c + b \|a +--R Type: Expression Integer +--E + +--S 13 +cc2:=aa.2-bb1 +--R +--R (6) +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (2b\|- a \|c \|a x + b x + c + (- b x - 2b c)\|- a ) +--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 2 | 2 +--R (- 4b\|- 4a c + b \|c \|a x + b x + c + (2b x + 4b c)\|- 4a c + b ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +-----------+ +--R 2 | 2 +---+ +-+ +--R - 2log(a x + b x + c)\|- 4a c + b \|- a \|c +--R + +--R +-----------+ +--R | 2 +---+ +--R - 2b x\|- 4a c + b \|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +---+ +-+ +--R (4a x + 2b x)\|- 4a c + b \|- a \|c +--R + +--R +-----------+ +--R 2 | 2 +---+ +--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|- a +--R / +--R +-----------+ +--------------+ +--R | 2 +---+ +-+ | 2 +--R 4a\|- 4a c + b \|- a \|c \|a x + b x + c +--R + +--R +-----------+ +--R | 2 +---+ +--R (- 2a b x - 4a c)\|- 4a c + b \|- a +--R Type: Expression Integer +--E + +--S 14 +cc3:=aa.2-bb1 +--R +--R (7) +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (2b\|- a \|c \|a x + b x + c + (- b x - 2b c)\|- a ) +--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 2 | 2 +--R (- 4b\|- 4a c + b \|c \|a x + b x + c + (2b x + 4b c)\|- 4a c + b ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +-----------+ +--R 2 | 2 +---+ +-+ +--R - 2log(a x + b x + c)\|- 4a c + b \|- a \|c +--R + +--R +-----------+ +--R | 2 +---+ +--R - 2b x\|- 4a c + b \|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +---+ +-+ +--R (4a x + 2b x)\|- 4a c + b \|- a \|c +--R + +--R +-----------+ +--R 2 | 2 +---+ +--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|- a +--R / +--R +-----------+ +--------------+ +--R | 2 +---+ +-+ | 2 +--R 4a\|- 4a c + b \|- a \|c \|a x + b x + c +--R + +--R +-----------+ +--R | 2 +---+ +--R (- 2a b x - 4a c)\|- 4a c + b \|- a +--R Type: Expression Integer +--E + +--S 15 14:266 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (8) +--R +---------+ +--------------+ +---------+ +--R +-+ | 2 | 2 2 | 2 +--R (- 4b\|c \|4a c - b \|a x + b x + c + (2b x + 4b c)\|4a c - b ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (4b\|- a \|c \|a x + b x + c + (- 2b x - 4b c)\|- a ) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 +---+ +-+ +---+ | 2 +--R (- 2log(a x + b x + c)\|- a \|c - 2b x\|- a )\|4a c - b +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 +---+ +-+ 2 +---+ +--R ((4a x + 2b x)\|- a \|c + (b x + 2c)log(a x + b x + c)\|- a ) +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--------------+ +--R +---+ +-+ | 2 | 2 +--R 4a\|- a \|c \|4a c - b \|a x + b x + c +--R + +--R +---------+ +--R +---+ | 2 +--R (- 2a b x - 4a c)\|- a \|4a c - b +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.267~~~~~$\displaystyle @@ -127,7 +583,7 @@ $$ <<*>>= )clear all ---S 3 of 15 +--S 16 aa:=integrate(x^2/(a*x^2+b*x+c),x) --R --R @@ -173,6 +629,141 @@ aa:=integrate(x^2/(a*x^2+b*x+c),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 17 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--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 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 18 +bb1:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.1 +--R +--R +--R (3) +--R 2 +--R (- 2a c + b ) +--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 log(a x + b x + c) + 2a x)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R Type: Expression Integer +--E + +--S 19 +bb2:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2a \|4a c - b +--R Type: Expression Integer +--E + +--S 20 +cc1:=bb1-aa.1 +--R +--R (5) +--R 2 +--R (- 2a c + b ) +--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 (- 2a c + b ) +--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 2a \|- 4a c + b +--R Type: Expression Integer +--E + +--S 21 14:267 Schaums and Axiom differ by a constant +dd1:=complexNormalize cc1 +--R +--R 2 3 2 2 +--R (- 2a c + b )log(- 16a c + 4a b ) +--R (6) --------------------------------- +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R Type: Expression Integer +--E @ \section{\cite{1}:14.268~~~~~$\displaystyle @@ -184,7 +775,7 @@ $$ <<*>>= )clear all ---S 4 of 15 +--S 22 14:268 Axiom cannot compute this integral aa:=integrate(x^m/(a*x^2+b*x+c),x) --R --R @@ -195,6 +786,7 @@ aa:=integrate(x^m/(a*x^2+b*x+c),x) --I c + %N b + %N a --R Type: Union(Expression Integer,...) --E + @ \section{\cite{1}:14.269~~~~~$\displaystyle @@ -206,7 +798,7 @@ $$ <<*>>= )clear all ---S 5 of 15 +--S 23 aa:=integrate(1/(x*(a*x^2+b*x+c)),x) --R --R @@ -251,6 +843,179 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 24 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--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 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 25 +bb1:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.1 +--R +--R +--R (3) +--R - +--R b +--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 x | 2 +--R log(--------------)\|- 4a c + b +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 26 +bb2:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.2 +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R (2a x + b)\|4a c - b x | 2 +--R - 2b atan(----------------------) + log(--------------)\|4a c - b +--R 2 2 +--R 4a c - b a x + b x + c +--R (4) ------------------------------------------------------------------- +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R Type: Expression Integer +--E + +--S 27 +cc1:=bb1-aa.1 +--R +--R (5) +--R - +--R b +--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 b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b +--R + +--R 2 2 3 +--R (- 8a c + 2a b )x - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R 2 x | 2 +--R (log(a x + b x + c) - 2log(x) + log(--------------))\|- 4a c + b +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 28 +dd1:=expandLog cc1 +--R +--R (6) +--R - +--R b +--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 - +--R b +--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 2b log(a x + b x + c) +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 29 14:269 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R b log(- 16a c + 4a b ) +--R (7) - ---------------------- +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E @ \section{\cite{1}:14.270~~~~~$\displaystyle @@ -262,7 +1027,7 @@ $$ <<*>>= )clear all ---S 6 of 15 +--S 30 aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x) --R --R @@ -308,6 +1073,186 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 31 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--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 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 32 +bb1:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.1 +--R +--R +--R (3) +--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 2 +-----------+ +--R a x + b x + c | 2 +--R (b x log(--------------) - 2c)\|- 4a c + b +--R 2 +--R x +--R / +--R +-----------+ +--R 2 | 2 +--R 2c x\|- 4a c + b +--R Type: Expression Integer +--E + +--S 33 +bb2:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.2 +--R +--R +--R (4) +--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 2 +---------+ +--R a x + b x + c | 2 +--R (b x log(--------------) - 2c)\|4a c - b +--R 2 +--R x +--R / +--R +---------+ +--R 2 | 2 +--R 2c x\|4a c - b +--R Type: Expression Integer +--E + +--S 34 +cc1:=bb1-aa.1 +--R +--R (5) +--R 2 +--R (- 2a c + b ) +--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 (- 2a c + b ) +--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 a x + b x + c | 2 +--R (- b log(a x + b x + c) + 2b log(x) + b log(--------------))\|- 4a c + b +--R 2 +--R x +--R / +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E + +--S 35 +dd1:=expandLog cc1 +--R +--R (6) +--R 2 +--R (- 2a c + b ) +--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 (- 2a c + b ) +--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 2 +--R (4a c - 2b )log(a x + b x + c) +--R / +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E + +--S 36 14:270 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 2 3 2 2 +--R (- 2a c + b )log(- 16a c + 4a b ) +--R (7) --------------------------------- +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E @ \section{\cite{1}:14.271~~~~~$\displaystyle @@ -320,7 +1265,7 @@ $$ <<*>>= )clear all ---S 7 of 15 +--S 37 14:271 Axiom cannot compute this integral aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x) --R --R @@ -331,6 +1276,7 @@ aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x) --I (c + %N b + %N a)%N --R Type: Union(Expression Integer,...) --E + @ \section{\cite{1}:14.272~~~~~$\displaystyle @@ -342,7 +1288,7 @@ $$ <<*>>= )clear all ---S 8 of 15 +--S 38 aa:=integrate(1/(a*x^2+b*x+c)^2,x) --R --R @@ -383,6 +1329,213 @@ aa:=integrate(1/(a*x^2+b*x+c)^2,x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 39 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--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 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 40 +bb1:=(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1 +--R +--R +--R (3) +--R 2 2 +--R (- a b x - b x - b 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 +-----------+ +--R | 2 +--R (b x + 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 41 +bb2:=(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 2 (2a x + b)\|4a c - b +--R (- 2a b x - 2b x - 2b c)atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R | 2 +--R (b x + 2c)\|4a c - b +--R / +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 42 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 2 2 +--R ((a b + 2a )x + (b + 2a b)x + (b + 2a)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 +-----------+ +--R | 2 +--R ((- b + 2a)x - 2c + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 43 +cc2:=aa.2-bb1 +--R +--R (6) +--R +---------+ +--R 2 2 | 2 +--R (a b x + b x + b c)\|4a c - b +--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 +--R 2 2 | 2 (2a x + b)\|4a c - b +--R (4a x + 4a b x + 4a c)\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +-----------+ +---------+ +--R | 2 | 2 +--R ((- b + 2a)x - 2c + b)\|- 4a c + b \|4a c - b +--R / +--R +-----------+ +---------+ +--R 2 2 2 3 2 2 | 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 44 +cc3:=aa.1-bb2 +--R +--R (7) +--R +---------+ +--R 2 2 | 2 +--R (2a x + 2a b x + 2a c)\|4a c - b +--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 +--R 2 2 | 2 (2a x + b)\|4a c - b +--R (2a b x + 2b x + 2b c)\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +-----------+ +---------+ +--R | 2 | 2 +--R ((- b + 2a)x - 2c + b)\|- 4a c + b \|4a c - b +--R / +--R +-----------+ +---------+ +--R 2 2 2 3 2 2 | 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 45 14:272 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (8) +--R +---------+ +--R | 2 +--R 2 2 2 (2a x + b)\|4a c - b +--R ((2a b + 4a )x + (2b + 4a b)x + (2b + 4a)c)atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R | 2 +--R ((- b + 2a)x - 2c + b)\|4a c - b +--R / +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.273~~~~~$\displaystyle @@ -394,7 +1547,7 @@ $$ <<*>>= )clear all ---S 9 of 15 +--S 46 aa:=integrate(x/(a*x^2+b*x+c)^2,x) --R --R @@ -440,6 +1593,177 @@ aa:=integrate(x/(a*x^2+b*x+c)^2,x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 47 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--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 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 48 +bb1:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1 +--R +--R +--R (3) +--R 2 2 +--R (- a b x - b x - b 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 +-----------+ +--R | 2 +--R (- b x - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 49 +bb2:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 2 (2a x + b)\|4a c - b +--R (- 2a b x - 2b x - 2b c)atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R | 2 +--R (- b x - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 50 +cc1:=bb1-aa.1 +--R +--R (5) +--R - +--R b +--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 b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b +--R + +--R 2 2 3 +--R (- 8a c + 2a b )x - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 51 +dd1:=expandLog cc1 +--R +--R (6) +--R - +--R b +--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 - +--R b +--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 2b log(a x + b x + c) +--R / +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 52 14:273 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R b log(- 16a c + 4a b ) +--R (7) - ------------------------- +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.274~~~~~$\displaystyle @@ -451,7 +1775,7 @@ $$ <<*>>= )clear all ---S 10 of 15 +--S 53 aa:=integrate(x^2/(a*x^2+b*x+c)^2,x) --R --R @@ -497,6 +1821,96 @@ aa:=integrate(x^2/(a*x^2+b*x+c)^2,x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 54 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--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 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 55 +bb1:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.1 +--R +--R (3) +--R 2 2 2 +--R (2a c x + 2a b c x + 2a 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 +-----------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|- 4a c + b +--R / +--R +-----------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 56 +bb2:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 2 2 (2a x + b)\|4a c - b +--R (4a c x + 4a b c x + 4a c )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|4a c - b +--R / +--R +---------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 57 14:274 Schaums and Axiom agree +cc1:=aa.1-bb1 +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.275~~~~~$\displaystyle @@ -514,7 +1928,7 @@ $$ <<*>>= )clear all ---S 11 of 15 +--S 58 14:275 Axiom cannot compute this integral aa:=integrate(x^m/(a*x^2+b*x+c)^n,x) --R --R @@ -525,6 +1939,7 @@ aa:=integrate(x^m/(a*x^2+b*x+c)^n,x) --I (c + %N b + %N a) --R Type: Union(Expression Integer,...) --E + @ \section{\cite{1}:14.276~~~~~$\displaystyle @@ -541,7 +1956,7 @@ $$ <<*>>= )clear all ---S 12 of 15 +--S 59 14:276 Axiom cannot compute this integral aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x) --R --R @@ -552,6 +1967,7 @@ aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x) --I (c + %N b + %N a) --R Type: Union(Expression Integer,...) --E + @ \section{\cite{1}:14.277~~~~~$\displaystyle @@ -568,7 +1984,7 @@ $$ <<*>>= )clear all ---S 13 of 15 +--S 60 aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x) --R --R @@ -641,6 +2057,380 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 61 +t1:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (2) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a 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 +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 62 +t2:=integrate(1/(x*(a*x^2+b*x+c)),x) +--R +--R +--R (3) +--R [ +--R b +--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 (- log(a x + b x + c) + 2log(x))\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R - 2b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|4a c - b +--R / +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 63 +bb1:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.1 +--R +--R +--R (4) +--R 2 2 2 2 +--R (- 2a b c x - 2a b c x - 2a b 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 3 2 2 4 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b 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 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 64 +bb2:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.1 +--R +--R +--R (5) +--R +---------+ +--R 2 3 2 2 4 2 3 | 2 +--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c)\|4a c - b +--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 2 2 | 2 +--R (- 4a b c x - 4a b c x - 4a b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 65 +bb3:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.2 +--R +--R +--R (6) +--R +---------+ +--R 2 2 2 2 | 2 +--R (- 2a b c x - 2a b c x - 2a b c )\|4a c - b +--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 3 2 2 4 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x - 8a b c + 2b c) +--R * +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R \|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 66 +bb4:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.2 +--R +--R +--R (7) +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R Type: Expression Integer +--E + +--S 67 +cc1:=aa.1-bb1 +--R +--R (8) +--R a b +--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 a b +--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 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 68 +dd1:=expandLog cc1 +--R +--R (9) +--R a b +--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 a b +--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 - 2a b log(a x + b x + c) +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 69 14:277 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R a b log(- 16a c + 4a b ) +--R (10) --------------------------- +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E @ \section{\cite{1}:14.278~~~~~$\displaystyle @@ -657,7 +2447,7 @@ $$ <<*>>= )clear all ---S 14 of 15 +--S 70 aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x) --R --R @@ -740,6 +2530,426 @@ aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 71 +t1:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (2) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a 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 +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 72 +t2:=integrate(1/(x*(a*x^2+b*x+c)^2),x) +--R +--R +--R (3) +--R [ +--R 2 3 2 2 4 2 3 +--R ((6a b c - a b )x + (6a b c - b )x + 6a b c - b 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 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R , +--R +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 73 +bb1:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.1 +--R +--R +--R (4) +--R 3 2 3 2 2 2 2 3 +--R (- 6a c x - 6a b c x - 6a c 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 2 2 4 3 3 5 2 2 2 4 +--R ((- 6a b c + a b )x + (- 6a b c + b )x + (- 6a b c + b c)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 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 74 +bb2:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.1 +--R +--R +--R (5) +--R 2 2 4 3 3 5 2 2 2 4 +--R ((- 6a b c + a b )x + (- 6a b c + b )x + (- 6a b c + b c)x) +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--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 3 2 3 2 2 2 2 3 | 2 +--R (- 12a c x - 12a b c x - 12a c x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 75 +bb3:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.2 +--R +--R +--R (6) +--R +---------+ +--R 3 2 3 2 2 2 2 3 | 2 +--R (- 6a c x - 6a b c x - 6a c x)\|4a c - b +--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 2 4 3 3 5 2 2 2 4 +--R ((12a b c - 2a b )x + (12a b c - 2b )x + (12a b c - 2b c)x) +--R * +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R \|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 76 +bb4:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.2 +--R +--R +--R (7) +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 12a c + 12a b c - 2a b )x + (- 12a b c + 12a b c - 2b )x +--R + +--R 2 3 2 2 4 +--R (- 12a c + 12a b c - 2b c)x +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|4a c - b +--R Type: Expression Integer +--E + +--S 77 +cc1:=aa.1-bb1 +--R +--R (8) +--R 2 +--R 6a +--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 6a +--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 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 78 +dd1:=expandLog cc1 +--R +--R (9) +--R 2 +--R 6a +--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 6a +--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 2 +--R - 12a log(a x + b x + c) +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 79 14:278 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 2 3 2 2 +--R 6a log(- 16a c + 4a b ) +--R (10) --------------------------- +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E @ \section{\cite{1}:14.279~~~~~$\displaystyle @@ -756,7 +2966,7 @@ $$ <<*>>= )clear all ---S 15 of 15 +--S 80 14:279 Axiom cannot compute this integral aa:=integrate(1/(x^m*(a*x^2+b*x+c)^n),x) --R --R