diff --git a/changelog b/changelog index 502aa1e..c10550c 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,11 @@ +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 +20080418 tpd src/input/schaum8.input fix Schaums-Axiom equivalence +20080418 tpd src/input/schaum7.input fix Schaums-Axiom equivalence +20080418 tpd src/input/schaum6.input fix Schaums-Axiom equivalence +20080418 tpd src/input/schaum2.input fix Schaums-Axiom equivalence +20080418 tpd src/input/schaum1.input fix Schaums-Axiom equivalence 20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence 20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence 20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence diff --git a/src/input/schaum1.input.pamphlet b/src/input/schaum1.input.pamphlet index 7e7e8c4..615d907 100644 --- a/src/input/schaum1.input.pamphlet +++ b/src/input/schaum1.input.pamphlet @@ -1201,7 +1201,7 @@ explog:=rule(%e^(n*log(x)) == x^n) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 94 14:80 Schaums and Axiom agree +--S 94 dd:=explog cc --R --R n + 1 n @@ -1210,8 +1210,14 @@ dd:=explog cc --R a n + a --R Type: Expression Integer --E + +--S 95 14:80 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ -The numerator is clearly zero but I cannot get Axiom to simplify it. \section{\cite{1}:14.81~~~~~$\displaystyle \int{x(ax+b)^n~dx}$} @@ -1221,7 +1227,7 @@ $$\int{x(ax+b)^n}= $$ <<*>>= )clear all ---S 95 +--S 96 aa:=integrate(x*(a*x+b)^n,x) --R --R 2 2 2 2 n log(a x + b) @@ -1232,7 +1238,7 @@ aa:=integrate(x*(a*x+b)^n,x) --R Type: Union(Expression Integer,...) --E ---S 96 +--S 97 bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2) --R --R n + 2 n + 1 @@ -1243,7 +1249,7 @@ bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2) --R Type: Expression Integer --E ---S 97 +--S 98 cc:=aa-bb --R --R (3) @@ -1258,7 +1264,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 98 +--S 99 explog:=rule(%e^(n*log(x)) == x^n) --R --R n log(x) n @@ -1266,7 +1272,7 @@ explog:=rule(%e^(n*log(x)) == x^n) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 99 14:81 Schaums and Axiom agreement cannot be determined +--S 100 dd:=explog cc --R --R (5) @@ -1280,6 +1286,13 @@ dd:=explog cc --R a n + 3a n + 2a --R Type: Expression Integer --E + +--S 101 +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.82~~~~~$\displaystyle \int{x^2(ax+b)^n~dx}$} @@ -1292,7 +1305,7 @@ $$ <<*>>= )clear all ---S 100 +--S 102 aa:=integrate(x^2*(a*x+b)^n,x) --R --R (1) @@ -1304,7 +1317,7 @@ aa:=integrate(x^2*(a*x+b)^n,x) --R Type: Union(Expression Integer,...) --E ---S 101 +--S 103 bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1))/((n+1)*a^3) --R --R (2) @@ -1319,7 +1332,7 @@ bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1) --R Type: Expression Integer --E ---S 102 14:82 Schaums and Axiom agreement cannot be determined +--S 104 cc:=aa-bb --R --R (3) @@ -1339,6 +1352,39 @@ cc:=aa-bb --R a n + 6a n + 11a n + 6a --R Type: Expression Integer --E + +--S 105 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 106 +dd:=explog cc +--R +--R (5) +--R 2 n + 3 2 n + 2 +--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b) +--R + +--R 2 2 2 2 n + 1 +--R (- b n - 5b n - 6b )(a x + b) +--R + +--R 3 2 3 3 3 2 2 2 2 2 3 n +--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Expression Integer +--E + +--S 107 14:82 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.83~~~~~$\displaystyle \int{x^m(ax+b)^n}~dx$} @@ -1361,7 +1407,7 @@ $$\int{x^m(ax+b)^n} $$ <<*>>= ---S 103 14:83 Axiom cannot do this integration +--S 108 14:83 Axiom cannot do this integration aa:=integrate(x^m*(a*x+b)^n,x) --R --R x diff --git a/src/input/schaum10.input.pamphlet b/src/input/schaum10.input.pamphlet index d0ba7d9..6149cc0 100644 --- a/src/input/schaum10.input.pamphlet +++ b/src/input/schaum10.input.pamphlet @@ -133,7 +133,7 @@ bb:=(x*sqrt(x^2-a^2))/2+a^2/2*log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 11 14:211 Axiom cannot simplify this expression +--S 11 cc:=aa-bb --R --R +-------+ +-------+ @@ -144,6 +144,16 @@ cc:=aa-bb --R Type: Expression Integer --E +--S 12 14:211 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(- a ) +--R (4) - ----------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.212~~~~~$\displaystyle @@ -154,7 +164,7 @@ $$ <<*>>= )clear all ---S 12 +--S 13 aa:=integrate(x^3/sqrt(x^2-a^2),x) --R --R @@ -168,7 +178,7 @@ aa:=integrate(x^3/sqrt(x^2-a^2),x) --R Type: Union(Expression Integer,...) --E ---S 13 +--S 14 bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2) --R --R +-------+ @@ -179,7 +189,7 @@ bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2) --R Type: Expression Integer --E ---S 14 14:212 Schaums and Axiom agree +--S 15 14:212 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -194,7 +204,7 @@ $$ <<*>>= )clear all ---S 15 +--S 16 aa:=integrate(1/(x*sqrt(x^2-a^2)),x) --R --R @@ -208,29 +218,134 @@ aa:=integrate(1/(x*sqrt(x^2-a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 16 -bb:=1/a*asec(abs(x/a)) +--S 17 +bb:=1/a*asec(x/a) --R ---R x ---R asec(abs(-)) ---R a ---R (2) ------------ ---R a +--R x +--R asec(-) +--R a +--R (2) ------- +--R a --R Type: Expression Integer --E ---S 17 14:213 Axiom cannot simplify this expression +--S 18 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R 2atan(--------------) - asec(abs(-)) ---R a a ---R (3) ------------------------------------ ---R a +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R a --R Type: Expression Integer --E + +--S 19 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 20 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x \|x - a - x +--R - 2%i log(------------------) + 4atan(--------------) - %pi +--R x a +--R (5) ----------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 21 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 22 +ee:=atanrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 23 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 24 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 25 14:213 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 2a +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.214~~~~~$\displaystyle @@ -241,7 +356,7 @@ $$ <<*>>= )clear all ---S 18 +--S 26 aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x) --R --R @@ -253,7 +368,7 @@ aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 19 +--S 27 bb:=sqrt(x^2-a^2)/(a^2*x) --R --R +-------+ @@ -265,7 +380,7 @@ bb:=sqrt(x^2-a^2)/(a^2*x) --R Type: Expression Integer --E ---S 20 14:214 Schaums and Axiom differ by a constant +--S 28 14:214 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 1 @@ -284,7 +399,7 @@ $$ <<*>>= )clear all ---S 21 +--S 29 aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) --R --R @@ -305,33 +420,143 @@ aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 22 -bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(abs(x/a)) +--S 30 +bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(x/a) --R ---R +-------+ ---R 2 x | 2 2 ---R x asec(abs(-)) + a\|x - a ---R a ---R (2) ---------------------------- ---R 3 2 ---R 2a x +--R +-------+ +--R | 2 2 2 x +--R a\|x - a + x asec(-) +--R a +--R (2) ----------------------- +--R 3 2 +--R 2a x --R Type: Expression Integer --E ---S 23 14:215 Axiom cannot simplify this expression +--S 31 cc:=aa-bb +--R --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R 2atan(--------------) - asec(abs(-)) ---R a a ---R (3) ------------------------------------ ---R 3 ---R 2a +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R 3 +--R 2a --R Type: Expression Integer --E +--S 32 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 33 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R - %i log(-----------------------) - asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 34 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 35 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 36 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 37 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 38 14:215 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + @ \section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$} @@ -341,7 +566,7 @@ $$ <<*>>= )clear all ---S 24 +--S 39 aa:=integrate(sqrt(x^2-a^2),x) --R --R @@ -360,7 +585,7 @@ aa:=integrate(sqrt(x^2-a^2),x) --R Type: Union(Expression Integer,...) --E ---S 25 +--S 40 bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -371,7 +596,7 @@ bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 26 14:216 Axiom cannot simplify this expression +--S 41 cc:=aa-bb --R --R +-------+ +-------+ @@ -382,6 +607,15 @@ cc:=aa-bb --R Type: Expression Integer --E +--S 42 14:216 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(- a ) +--R (4) ----------- +--R 2 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$} @@ -391,7 +625,7 @@ $$ <<*>>= )clear all ---S 27 +--S 43 aa:=integrate(x*sqrt(x^2-a^2),x) --R --R @@ -405,7 +639,7 @@ aa:=integrate(x*sqrt(x^2-a^2),x) --R Type: Union(Expression Integer,...) --E ---S 28 +--S 44 bb:=(x^2-a^2)^(3/2)/3 --R --R +-------+ @@ -416,7 +650,7 @@ bb:=(x^2-a^2)^(3/2)/3 --R Type: Expression Integer --E ---S 29 14:217 Schaums and Axiom agree +--S 45 14:217 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -433,7 +667,7 @@ $$ <<*>>= )clear all ---S 30 +--S 46 aa:=integrate(x^2*sqrt(x^2-a^2),x) --R --R @@ -452,7 +686,7 @@ aa:=integrate(x^2*sqrt(x^2-a^2),x) --R Type: Union(Expression Integer,...) --E ---S 31 +--S 47 bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -463,7 +697,7 @@ bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 32 14:218 Axiom cannot simplify this expression +--S 48 cc:=aa-bb --R --R +-------+ +-------+ @@ -473,6 +707,16 @@ cc:=aa-bb --R 8 --R Type: Expression Integer --E + +--S 49 14:218 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R a log(- a ) +--R (4) ----------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.219~~~~~$\displaystyle @@ -483,7 +727,7 @@ $$ <<*>>= )clear all ---S 33 +--S 50 aa:=integrate(x^3*sqrt(x^2-a^2),x) --R --R @@ -501,7 +745,7 @@ aa:=integrate(x^3*sqrt(x^2-a^2),x) --R Type: Union(Expression Integer,...) --E ---S 34 +--S 51 bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3 --R --R +-------+ @@ -512,7 +756,7 @@ bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3 --R Type: Expression Integer --E ---S 35 14:219 Schaums and Axiom agree +--S 52 14:219 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -528,7 +772,7 @@ $$ <<*>>= )clear all ---S 36 +--S 53 aa:=integrate(sqrt(x^2-a^2)/x,x) --R --R @@ -544,26 +788,129 @@ aa:=integrate(sqrt(x^2-a^2)/x,x) --R Type: Union(Expression Integer,...) --E ---S 37 -bb:=sqrt(x^2-a^2)-a*asec(abs(x/a)) +--S 54 +bb:=sqrt(x^2-a^2)-a*asec(x/a) --R ---R +-------+ ---R x | 2 2 ---R (2) - a asec(abs(-)) + \|x - a ---R a +--R +-------+ +--R | 2 2 x +--R (2) \|x - a - a asec(-) +--R a --R Type: Expression Integer --E ---S 38 14:220 Axiom cannot simplify this expression +--S 55 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R (3) - 2a atan(--------------) + a asec(abs(-)) ---R a a +--R \|x - a - x x +--R (3) - 2a atan(--------------) + a asec(-) +--R a a --R Type: Expression Integer --E + +--S 56 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 57 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R (5) %i a log(-----------------------) + a asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R Type: Expression Complex Integer +--E + +--S 58 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 59 +ee:=asecrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 2%i a log(------------------) + 2%i a log(-----------------------) + a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R -------------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 60 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i a log(\|x - a - x + %i a) + 2%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 2%i a log(x |------- + %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi +--R | 2 +--R \| x +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 61 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i a log(\|x - a + %i a) - 2%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 2%i a log(\|x - a - x - %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 62 14:220 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R a %pi +--R (10) ----- +--R 2 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.221~~~~~$\displaystyle @@ -574,7 +921,7 @@ $$ <<*>>= )clear all ---S 39 +--S 63 aa:=integrate(sqrt(x^2-a^2)/x^2,x) --R --R @@ -588,7 +935,7 @@ aa:=integrate(sqrt(x^2-a^2)/x^2,x) --R Type: Union(Expression Integer,...) --E ---S 40 +--S 64 bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -599,7 +946,7 @@ bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 41 14:221 Axiom cannot simplify this expression +--S 65 cc:=aa-bb --R --R +-------+ +-------+ @@ -608,6 +955,14 @@ cc:=aa-bb --R Type: Expression Integer --E +--S 66 14:221 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(- a ) - 1 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.222~~~~~$\displaystyle @@ -619,7 +974,7 @@ $$ <<*>>= )clear all ---S 42 +--S 67 aa:=integrate(sqrt(x^2-a^2)/x^3,x) --R --R @@ -640,31 +995,136 @@ aa:=integrate(sqrt(x^2-a^2)/x^3,x) --R Type: Union(Expression Integer,...) --E ---S 43 -bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(abs(x/a)) +--S 68 +bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(x/a) --R ---R +-------+ ---R 2 x | 2 2 ---R x asec(abs(-)) - a\|x - a ---R a ---R (2) ---------------------------- ---R 2 ---R 2a x +--R +-------+ +--R | 2 2 2 x +--R - a\|x - a + x asec(-) +--R a +--R (2) ------------------------- +--R 2 +--R 2a x --R Type: Expression Integer --E ---S 44 14:222 Axiom cannot simplify this expression +--S 69 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R 2atan(--------------) - asec(abs(-)) ---R a a ---R (3) ------------------------------------ ---R 2a +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R 2a --R Type: Expression Integer --E + +--S 70 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 71 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x \|x - a - x +--R - 2%i log(------------------) + 4atan(--------------) - %pi +--R x a +--R (5) ----------------------------------------------------------- +--R 4a +--R Type: Expression Complex Integer +--E + +--S 72 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 73 +ee:=atanrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 4a +--R Type: Expression Complex Integer +--E + +--S 74 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 4a +--R Type: Expression Complex Integer +--E + +--S 75 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 4a +--R Type: Expression Complex Integer +--E + +--S 76 14:222 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 4a +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$} @@ -674,7 +1134,7 @@ $$ <<*>>= )clear all ---S 45 +--S 77 aa:=integrate(1/(x^2-a^2)^(3/2),x) --R --R @@ -686,7 +1146,7 @@ aa:=integrate(1/(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 46 +--S 78 bb:=-x/(a^2*sqrt(x^2-a^2)) --R --R x @@ -697,7 +1157,7 @@ bb:=-x/(a^2*sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 47 14:223 Schaums and Axiom differ by a constant +--S 79 14:223 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 1 @@ -716,7 +1176,7 @@ $$ <<*>>= )clear all ---S 48 +--S 80 aa:=integrate(x/(x^2-a^2)^(3/2),x) --R --R @@ -730,7 +1190,7 @@ aa:=integrate(x/(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 49 +--S 81 bb:=-1/sqrt(x^2-a^2) --R --R 1 @@ -741,7 +1201,7 @@ bb:=-1/sqrt(x^2-a^2) --R Type: Expression Integer --E ---S 50 14:224 Schaums and Axiom agree +--S 82 14:224 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -757,7 +1217,7 @@ $$ <<*>>= )clear all ---S 51 +--S 83 aa:=integrate(x^2/(x^2-a^2)^(3/2),x) --R --R @@ -771,7 +1231,7 @@ aa:=integrate(x^2/(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 52 +--S 84 bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -784,7 +1244,7 @@ bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 53 14:225 Axiom cannot simplify this expression +--S 85 cc:=aa-bb --R --R +-------+ +-------+ @@ -792,6 +1252,14 @@ cc:=aa-bb --R (3) - log(\|x - a + x) - log(\|x - a - x) - 1 --R Type: Expression Integer --E + +--S 86 14:225 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(- a ) - 1 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.226~~~~~$\displaystyle @@ -802,7 +1270,7 @@ $$ <<*>>= )clear all ---S 54 +--S 87 aa:=integrate(x^3/(x^2-a^2)^(3/2),x) --R --R @@ -816,7 +1284,7 @@ aa:=integrate(x^3/(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 55 +--S 88 bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2) --R --R 2 2 @@ -828,7 +1296,7 @@ bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2) --R Type: Expression Integer --E ---S 56 14:226 Schaums and Axiom agree +--S 89 14:226 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -845,7 +1313,7 @@ $$ <<*>>= )clear all ---S 57 +--S 90 aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) --R --R @@ -861,33 +1329,143 @@ aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 58 -bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(abs(x/a)) +--S 91 +bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(x/a) --R ---R +-------+ ---R | 2 2 x ---R - \|x - a asec(abs(-)) - a ---R a ---R (2) ---------------------------- ---R +-------+ ---R 3 | 2 2 ---R a \|x - a +--R +-------+ +--R x | 2 2 +--R - asec(-)\|x - a - a +--R a +--R (2) ----------------------- +--R +-------+ +--R 3 | 2 2 +--R a \|x - a --R Type: Expression Integer --E ---S 59 14:227 Axiom cannot simplify this expression +--S 92 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R - 2atan(--------------) + asec(abs(-)) ---R a a ---R (3) -------------------------------------- ---R 3 ---R a +--R \|x - a - x x +--R - 2atan(--------------) + asec(-) +--R a a +--R (3) --------------------------------- +--R 3 +--R a --R Type: Expression Integer --E + +--S 93 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 94 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R %i log(-----------------------) + asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ----------------------------------------- +--R 3 +--R a +--R Type: Expression Complex Integer +--E + +--S 95 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 96 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 2%i log(------------------) + 2%i log(-----------------------) + %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) -------------------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 97 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a - x + %i a) + 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 2%i log(x |------- + %i a) - 2%i log(x) + 2%i log(- 1) + %pi +--R | 2 +--R \| x +--R / +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 98 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a + %i a) - 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 2%i log(\|x - a - x - %i a) - 2%i log(x) + 2%i log(- 1) + %pi +--R / +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 99 14:227 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) --- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + @ \section{\cite{1}:14.228~~~~~$\displaystyle @@ -898,7 +1476,7 @@ $$ <<*>>= )clear all ---S 60 +--S 100 aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x) --R --R @@ -910,7 +1488,7 @@ aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 61 +--S 101 bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2)) --R --R 2 2 @@ -922,7 +1500,7 @@ bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 62 14:228 Schaums and Axiom differ by a constant +--S 102 14:228 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 2 @@ -944,7 +1522,7 @@ $$ <<*>>= )clear all ---S 63 +--S 103 aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) --R --R @@ -965,35 +1543,143 @@ aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 64 -bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(abs(x/a)) +--S 104 +bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(x/a) --R ---R +-------+ ---R 2 | 2 2 x 2 3 ---R - 3x \|x - a asec(abs(-)) - 3a x + a ---R a ---R (2) ---------------------------------------- ---R +-------+ ---R 5 2 | 2 2 ---R 2a x \|x - a +--R +-------+ +--R 2 x | 2 2 2 3 +--R - 3x asec(-)\|x - a - 3a x + a +--R a +--R (2) ----------------------------------- +--R +-------+ +--R 5 2 | 2 2 +--R 2a x \|x - a --R Type: Expression Integer --E ---S 65 14:229 Axiom cannot simplify this expression +--S 105 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R - 6atan(--------------) + 3asec(abs(-)) ---R a a ---R (3) --------------------------------------- ---R 5 ---R 2a +--R \|x - a - x x +--R - 6atan(--------------) + 3asec(-) +--R a a +--R (3) ---------------------------------- +--R 5 +--R 2a --R Type: Expression Integer --E -@ +--S 106 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 107 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R 3%i log(-----------------------) + 3asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 108 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 109 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 6%i log(------------------) + 6%i log(-----------------------) + 3%pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) --------------------------------------------------------------------- +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 110 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 6%i log(\|x - a - x + %i a) + 6%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 6%i log(x |------- + %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi +--R | 2 +--R \| x +--R / +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 111 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 6%i log(\|x - a + %i a) - 6%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 6%i log(\|x - a - x - %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi +--R / +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 112 14:229 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3%pi +--R (10) ---- +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E +@ \section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$} $$\int{(x^2-a^2)^{3/2}}= \frac{x(x^2-a^2)^{3/2}}{4}-\frac{3a^2x\sqrt{x^2-a^2}}{8}+ @@ -1002,7 +1688,7 @@ $$ <<*>>= )clear all ---S 66 +--S 113 aa:=integrate((x^2-a^2)^(3/2),x) --R --R @@ -1024,7 +1710,7 @@ aa:=integrate((x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 67 +--S 114 bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -1035,7 +1721,7 @@ bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 68 14:230 Axiom cannot simplify this expression +--S 115 cc:=aa-bb --R --R +-------+ +-------+ @@ -1046,6 +1732,16 @@ cc:=aa-bb --R Type: Expression Integer --E +--S 116 14:230 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R 3a log(- a ) +--R (4) - ------------ +--R 8 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$} @@ -1053,7 +1749,7 @@ $$\int{x(x^2-a^2)^{3/2}}=\frac{(x^2-a^2)^{5/2}}{5}$$ <<*>>= )clear all ---S 69 +--S 117 aa:=integrate(x*(x^2-a^2)^(3/2),x) --R --R @@ -1071,7 +1767,7 @@ aa:=integrate(x*(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 70 +--S 118 bb:=(x^2-a^2)^(5/2)/5 --R --R +-------+ @@ -1082,7 +1778,7 @@ bb:=(x^2-a^2)^(5/2)/5 --R Type: Expression Integer --E ---S 71 14:231 Schaums and Axiom agree +--S 119 14:231 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -1099,7 +1795,7 @@ $$ <<*>>= )clear all ---S 72 +--S 120 aa:=integrate(x^2*(x^2-a^2)^(3/2),x) --R --R @@ -1128,7 +1824,7 @@ aa:=integrate(x^2*(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 73 +--S 121 bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -1139,7 +1835,7 @@ bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^ --R Type: Expression Integer --E ---S 74 14:232 Axiom cannot simplify this expression +--S 122 cc:=aa-bb --R --R +-------+ +-------+ @@ -1149,6 +1845,16 @@ cc:=aa-bb --R 16 --R Type: Expression Integer --E + +--S 123 14:232 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 6 2 +--R a log(- a ) +--R (4) - ----------- +--R 16 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$} @@ -1158,7 +1864,7 @@ $$ <<*>>= )clear all ---S 75 +--S 124 aa:=integrate(x^3*(x^2-a^2)^(3/2),x) --R --R @@ -1188,7 +1894,7 @@ aa:=integrate(x^3*(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 76 +--S 125 bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5 --R --R +-------+ @@ -1199,7 +1905,7 @@ bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5 --R Type: Expression Integer --E ---S 77 14:233 Schaums and Axiom agree +--S 126 14:233 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -1216,7 +1922,7 @@ $$ <<*>>= )clear all ---S 78 +--S 127 aa:=integrate((x^2-a^2)^(3/2)/x,x) --R --R @@ -1237,28 +1943,135 @@ aa:=integrate((x^2-a^2)^(3/2)/x,x) --R Type: Union(Expression Integer,...) --E ---S 79 -bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(abs(x/a)) +--S 128 +bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(x/a) --R ---R +-------+ ---R 3 x 2 2 | 2 2 ---R 3a asec(abs(-)) + (x - 4a )\|x - a ---R a ---R (2) -------------------------------------- ---R 3 +--R +-------+ +--R 2 2 | 2 2 3 x +--R (x - 4a )\|x - a + 3a asec(-) +--R a +--R (2) --------------------------------- +--R 3 --R Type: Expression Integer --E ---S 80 14:234 Axiom cannot simplify this expression +--S 129 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R 3 \|x - a - x 3 x ---R (3) 2a atan(--------------) - a asec(abs(-)) ---R a a +--R 3 \|x - a - x 3 x +--R (3) 2a atan(--------------) - a asec(-) +--R a a --R Type: Expression Integer --E + +--S 130 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 131 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R 3 \| x 3 \|x - a - x 3 +--R - 2%i a log(------------------) + 4a atan(--------------) - a %pi +--R x a +--R (5) ----------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 132 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 133 +ee:=atanrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R 3 \| x 3 - \|x - a + x + %i a 3 +--R - 2%i a log(------------------) - 2%i a log(-----------------------) - a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R ---------------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 134 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R 2%i a log(\|x - a - x + %i a) - 2%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R 3 |x - a 3 3 3 +--R - 2%i a log(x |------- + %i a) + 2%i a log(x) - 2%i a log(- 1) - a %pi +--R | 2 +--R \| x +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 135 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R - 2%i a log(\|x - a + %i a) + 2%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R 3 | 2 2 3 3 3 +--R - 2%i a log(\|x - a - x - %i a) + 2%i a log(x) - 2%i a log(- 1) - a %pi +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 136 14:234 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3 +--R a %pi +--R (10) - ----- +--R 2 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.235~~~~~$\displaystyle @@ -1270,7 +2083,7 @@ $$ <<*>>= )clear all ---S 81 +--S 137 aa:=integrate((x^2-a^2)^{3/2}/x^2,x) --R --R @@ -1289,7 +2102,7 @@ aa:=integrate((x^2-a^2)^{3/2}/x^2,x) --R Type: Union(Expression Integer,...) --E ---S 82 +--S 138 bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -1300,7 +2113,7 @@ bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 83 14:235 Axiom cannot simplify this expression +--S 139 cc:=aa-bb --R --R +-------+ +-------+ @@ -1310,6 +2123,17 @@ cc:=aa-bb --R 2 --R Type: Expression Integer --E + +--S 140 14:235 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 2 +--R 3a log(- a ) + 2a +--R (4) ------------------ +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.236~~~~~$\displaystyle @@ -1321,7 +2145,7 @@ $$ <<*>>= )clear all ---S 84 +--S 141 aa:=integrate((x^2-a^2)^(3/2)/x^3,x) --R --R @@ -1342,31 +2166,137 @@ aa:=integrate((x^2-a^2)^(3/2)/x^3,x) --R Type: Union(Expression Integer,...) --E ---S 85 -bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(abs(x/a)) +--S 142 +bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(x/a) --R ---R +-------+ ---R 2 x 2 2 | 2 2 ---R - 3a x asec(abs(-)) + (2x + a )\|x - a ---R a ---R (2) ------------------------------------------ ---R 2 ---R 2x +--R +-------+ +--R 2 2 | 2 2 2 x +--R (2x + a )\|x - a - 3a x asec(-) +--R a +--R (2) ----------------------------------- +--R 2 +--R 2x --R Type: Expression Integer --E ---S 86 14:236 Axiom cannot simplify this expression +--S 143 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R - 6a atan(--------------) + 3a asec(abs(-)) ---R a a ---R (3) ------------------------------------------- ---R 2 +--R \|x - a - x x +--R - 6a atan(--------------) + 3a asec(-) +--R a a +--R (3) -------------------------------------- +--R 2 --R Type: Expression Integer --E + +--S 144 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 145 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R 3%i a log(-----------------------) + 3a asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ----------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 146 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 147 +ee:=asecrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 6%i a log(------------------) + 6%i a log(-----------------------) + 3a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R --------------------------------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 148 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 6%i a log(\|x - a - x + %i a) + 6%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 6%i a log(x |------- + %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a %pi +--R | 2 +--R \| x +--R / +--R 4 +--R Type: Expression Complex Integer +--E + +--S 149 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 6%i a log(\|x - a + %i a) - 6%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 6%i a log(\|x - a - x - %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a %pi +--R / +--R 4 +--R Type: Expression Complex Integer +--E + +--S 150 14:236 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3a %pi +--R (10) ------ +--R 4 +--R Type: Expression Complex Integer +--E + )spool )lisp (bye) @ diff --git a/src/input/schaum11.input.pamphlet b/src/input/schaum11.input.pamphlet index 6cd59a6..fc3117e 100644 --- a/src/input/schaum11.input.pamphlet +++ b/src/input/schaum11.input.pamphlet @@ -15,7 +15,7 @@ $$\int{\frac{1}{\sqrt{a^2-x^2}}}=\ln\left(x+\sqrt{a^2-x^2}\right)$$ )set message auto off )clear all ---S 1 of 28 +--S 1 aa:=integrate(1/(sqrt(a^2-x^2)),x) --R --R @@ -26,6 +26,96 @@ aa:=integrate(1/(sqrt(a^2-x^2)),x) --R x --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=asin(x/a) +--R +--R x +--R (2) asin(-) +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) - 2atan(----------------) - asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 4 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 5 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R - \|- x + a + %i x + a x +--R (5) %i log(-------------------------) - asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 6 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 7 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) - %i log(--------------------) + %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 8 +ff:=rootSimp ee +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i\|x - a - %i x - \|x - a + x - %i a +--R (8) - %i log(-------------------) + %i log(-----------------------) +--R a +-------+ +--R | 2 2 +--R \|x - a + x + %i a +--R Type: Expression Complex Integer +--E + +--S 9 14:238 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (9) 0 +--R Type: Expression Complex Integer +--E + @ \section{\cite{1}:14.238~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{a^2-x^2}}}$} @@ -33,7 +123,7 @@ $$\int{\frac{x}{\sqrt{a^2-x^2}}}=\sqrt{a^2-x^2}$$ <<*>>= )clear all ---S 2 of 28 +--S 10 aa:=integrate(x/(sqrt(a^2-x^2)),x) --R --R @@ -45,6 +135,22 @@ aa:=integrate(x/(sqrt(a^2-x^2)),x) --R \|- x + a - a --R Type: Union(Expression Integer,...) --E + +--S 11 +bb:=-sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 +--R (2) - \|- x + a +--R Type: Expression Integer +--E + +--S 12 14:238 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R (3) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.239~~~~~$\displaystyle @@ -55,7 +161,7 @@ $$ <<*>>= )clear all ---S 3 of 28 +--S 13 aa:=integrate(x^2/sqrt(a^2-x^2),x) --R --R @@ -75,6 +181,127 @@ aa:=integrate(x^2/sqrt(a^2-x^2),x) --R 4a\|- x + a + 2x - 4a --R Type: Union(Expression Integer,...) --E + +--S 14 +bb:=-(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a) +--R +--R +---------+ +--R | 2 2 2 x +--R - x\|- x + a + a asin(-) +--R a +--R (2) --------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 16 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 17 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 2 - \|- x + a + %i x + a 2 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 18 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 19 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 20 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 21 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 22 14:239 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E + @ \section{\cite{1}:14.240~~~~~$\displaystyle @@ -85,7 +312,7 @@ $$ <<*>>= )clear all ---S 4 of 28 +--S 23 aa:=integrate(x^3/sqrt(a^2-x^2),x) --R --R @@ -98,6 +325,27 @@ aa:=integrate(x^3/sqrt(a^2-x^2),x) --R (3x - 12a )\|- x + a - 9a x + 12a --R Type: Union(Expression Integer,...) --E + +--S 24 +bb:=(a^2-x^2)^(3/2)/3-a^2*sqrt(a^2-x^2) +--R +--R +---------+ +--R 2 2 | 2 2 +--R (- x - 2a )\|- x + a +--R (2) ------------------------ +--R 3 +--R Type: Expression Integer +--E + +--S 25 14:240 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 3 +--R 2a +--R (3) - --- +--R 3 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.241~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{a^2-x^2}}}$} @@ -107,7 +355,7 @@ $$ <<*>>= )clear all ---S 5 of 28 +--S 26 aa:=integrate(1/(x*sqrt(a^2-x^2)),x) --R --R @@ -120,6 +368,66 @@ aa:=integrate(1/(x*sqrt(a^2-x^2)),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 27 +bb:=-1/a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a + a +--R log(----------------) +--R x +--R (2) - --------------------- +--R a +--R Type: Expression Integer +--E + +--S 28 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) +--R x x +--R (3) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 29 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) +--R (4) ------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 30 +ee:=complexNormalize dd +--R +--R x +--R 2log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 31 14:241 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 2log(\|- 1 ) +--R (6) ------------ +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.242~~~~~$\displaystyle @@ -130,7 +438,7 @@ $$ <<*>>= )clear all ---S 6 of 28 +--S 32 aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x) --R --R @@ -143,6 +451,25 @@ aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x) --R a x\|- x + a - a x --R Type: Union(Expression Integer,...) --E + +--S 33 +bb:=-sqrt(a^2-x^2)/(a^2*x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a +--R (2) - ------------ +--R 2 +--R a x +--R Type: Expression Integer +--E + +--S 34 14:242 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.243~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{a^2-x^2}}}$} @@ -153,7 +480,7 @@ $$ <<*>>= )clear all ---S 7 of 28 +--S 35 aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x) --R --R @@ -173,6 +500,71 @@ aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x) --R 4a x \|- x + a + 2a x - 4a x --R Type: Union(Expression Integer,...) --E + +--S 36 +bb:=-sqrt(a^2-x^2)/(2*a^2*x^2)-1/(2*a^3)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a | 2 2 +--R - x log(----------------) - a\|- x + a +--R x +--R (2) ----------------------------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 37 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) +--R x x +--R (3) --------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 38 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) +--R (4) ------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 39 +ee:=complexNormalize dd +--R +--R x +--R log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) - ------------ +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 40 14:243 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R log(\|- 1 ) +--R (6) ----------- +--R 3 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.244~~~~~$\displaystyle\int{\sqrt{a^2-x^2}}~dx$} @@ -182,7 +574,7 @@ $$ <<*>>= )clear all ---S 8 of 28 +--S 41 aa:=integrate(sqrt(a^2-x^2),x) --R --R @@ -202,6 +594,127 @@ aa:=integrate(sqrt(a^2-x^2),x) --R 4a\|- x + a + 2x - 4a --R Type: Union(Expression Integer,...) --E + +--S 42 +bb:=(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a) +--R +--R +---------+ +--R | 2 2 2 x +--R x\|- x + a + a asin(-) +--R a +--R (2) ------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 44 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 45 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 \|- x + a - a +--R - %i a log(--------------------) - 2a atan(----------------) +--R a x +--R (5) ------------------------------------------------------------ +--R 2 +--R Type: Expression Complex Integer +--E + +--S 46 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 47 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 48 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 49 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 50 14:244 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.245~~~~~$\displaystyle\int{x\sqrt{a^2-x^2}}~dx$} @@ -211,7 +724,7 @@ $$ <<*>>= )clear all ---S 9 of 28 +--S 51 aa:=integrate(x*sqrt(a^2-x^2),x) --R --R @@ -224,6 +737,27 @@ aa:=integrate(x*sqrt(a^2-x^2),x) --R (3x - 12a )\|- x + a - 9a x + 12a --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=-(a^2-x^2)^(3/2)/3 +--R +--R +---------+ +--R 2 2 | 2 2 +--R (x - a )\|- x + a +--R (2) --------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 53 14:245 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 3 +--R a +--R (3) - -- +--R 3 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.246~~~~~$\displaystyle @@ -235,7 +769,7 @@ $$ <<*>>= )clear all ---S 10 of 28 +--S 54 aa:=integrate(x^2*sqrt(a^2-x^2),x) --R --R @@ -259,6 +793,126 @@ aa:=integrate(x^2*sqrt(a^2-x^2),x) --R (32a x - 64a )\|- x + a + 8x - 64a x + 64a --R Type: Union(Expression Integer,...) --E + +--S 55 +bb:=-(x*(a^2-x^2)^(3/2))/4+(a^2*x*sqrt(a^2-x^2))/8+a^4/8*asin(x/a) +--R +--R +---------+ +--R 3 2 | 2 2 4 x +--R (2x - a x)\|- x + a + a asin(-) +--R a +--R (2) ----------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 4 \|- x + a - a 4 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 57 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 58 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 4 - \|- x + a + %i x + a 4 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 59 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 60 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 61 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 4 |- x + a 4 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 4 | 2 2 4 4 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 62 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 4 | 2 2 4 4 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 63 14:246 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.247~~~~~$\displaystyle @@ -269,7 +923,7 @@ $$ <<*>>= )clear all ---S 11 of 28 +--S 64 aa:=integrate(x^3*sqrt(a^2-x^2),x) --R --R @@ -283,6 +937,27 @@ aa:=integrate(x^3*sqrt(a^2-x^2),x) --R (15x - 180a x + 240a )\|- x + a - 75a x + 300a x - 240a --R Type: Union(Expression Integer,...) --E + +--S 65 +bb:=(a^2-x^2)^(5/2)/5-(a^2*(a^2-x^2)^(3/2))/3 +--R +--R +---------+ +--R 4 2 2 4 | 2 2 +--R (3x - a x - 2a )\|- x + a +--R (2) ------------------------------ +--R 15 +--R Type: Expression Integer +--E + +--S 66 14:247 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 5 +--R 2a +--R (3) - --- +--R 15 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.248~~~~~$\displaystyle @@ -293,7 +968,7 @@ $$ <<*>>= )clear all ---S 12 of 28 +--S 67 aa:=integrate(sqrt(a^2-x^2)/x,x) --R --R @@ -308,6 +983,56 @@ aa:=integrate(sqrt(a^2-x^2)/x,x) --R \|- x + a - a --R Type: Union(Expression Integer,...) --E + +--S 68 +bb:=sqrt(a^2-x^2)-a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R \|- x + a + a | 2 2 +--R (2) - a log(----------------) + \|- x + a +--R x +--R Type: Expression Integer +--E + +--S 69 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R (3) a log(----------------) + a log(----------------) + a +--R x x +--R Type: Expression Integer +--E + +--S 70 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R (4) a log(\|- x + a + a) + a log(\|- x + a - a) - 2a log(x) + a +--R Type: Expression Integer +--E + +--S 71 +ee:=complexNormalize dd +--R +--R x +--R (5) - 2a log(-------) + a +--R +----+ +--R | 2 +--R \|- x +--R Type: Expression Integer +--E + +--S 72 14:248 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R (6) 2a log(\|- 1 ) + a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.249~~~~~$\displaystyle @@ -318,7 +1043,7 @@ $$ <<*>>= )clear all ---S 13 of 28 +--S 73 aa:=integrate(sqrt(a^2-x^2)/x^2,x) --R --R @@ -334,6 +1059,117 @@ aa:=integrate(sqrt(a^2-x^2)/x^2,x) --R x\|- x + a - a x --R Type: Union(Expression Integer,...) --E + +--S 74 +bb:=-sqrt(a^2-x^2)/x-asin(x/a) +--R +--R +---------+ +--R | 2 2 x +--R - \|- x + a - x asin(-) +--R a +--R (2) -------------------------- +--R x +--R Type: Expression Integer +--E + +--S 75 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) 2atan(----------------) + asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 76 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 77 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a \|- x + a - a +--R (5) %i log(--------------------) + 2atan(----------------) +--R a x +--R Type: Expression Complex Integer +--E + +--S 78 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 79 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) %i log(--------------------) - %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 80 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R |- x + a | 2 2 +--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R | 2 2 +--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 81 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 82 14:249 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.250~~~~~$\displaystyle @@ -345,7 +1181,7 @@ $$ <<*>>= )clear all ---S 14 of 28 +--S 83 aa:=integrate(sqrt(a^2-x^2)/x^3,x) --R --R @@ -365,6 +1201,67 @@ aa:=integrate(sqrt(a^2-x^2)/x^3,x) --R 4a x \|- x + a + 2a x - 4a x --R Type: Union(Expression Integer,...) --E + +--S 84 +bb:=-sqrt(a^2-x^2)/(2*x^2)+1/(2*a)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a | 2 2 +--R x log(----------------) - a\|- x + a +--R x +--R (2) --------------------------------------- +--R 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 85 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R - log(----------------) - log(----------------) +--R x x +--R (3) ----------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 86 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - log(\|- x + a + a) - log(\|- x + a - a) + 2log(x) +--R (4) --------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 87 +ee:=complexNormalize dd +--R +--R x +--R log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------ +--R a +--R Type: Expression Integer +--E + +--S 88 14:250 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R log(\|- 1 ) +--R (6) - ----------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.251~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^{3/2}}}$} @@ -374,7 +1271,7 @@ $$ <<*>>= )clear all ---S 15 of 28 +--S 89 aa:=integrate(1/(a^2-x^2)^(3/2),x) --R --R @@ -387,6 +1284,24 @@ aa:=integrate(1/(a^2-x^2)^(3/2),x) --R a \|- x + a + a x - a --R Type: Union(Expression Integer,...) --E + +--S 90 +bb:=x/(a^2*sqrt(a^2-x^2)) +--R +--R x +--R (2) -------------- +--R +---------+ +--R 2 | 2 2 +--R a \|- x + a +--R Type: Expression Integer +--E + +--S 91 14:251 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.252~~~~~$\displaystyle @@ -397,7 +1312,7 @@ $$ <<*>>= )clear all ---S 16 of 28 +--S 92 aa:=integrate(x/(a^2-x^2)^(3/2),x) --R --R @@ -409,6 +1324,26 @@ aa:=integrate(x/(a^2-x^2)^(3/2),x) --R a \|- x + a + a x - a --R Type: Union(Expression Integer,...) --E + +--S 93 +bb:=1/sqrt(a^2-x^2) +--R +--R 1 +--R (2) ------------ +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 94 14:252 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.253~~~~~$\displaystyle @@ -419,7 +1354,7 @@ $$ <<*>>= )clear all ---S 17 of 28 +--S 95 aa:=integrate(x^2/(a^2-x^2)^(3/2),x) --R --R @@ -435,6 +1370,118 @@ aa:=integrate(x^2/(a^2-x^2)^(3/2),x) --R a\|- x + a + x - a --R Type: Union(Expression Integer,...) --E + +--S 96 +bb:=x/sqrt(a^2-x^2)-asin(x/a) +--R +--R +---------+ +--R x | 2 2 +--R - asin(-)\|- x + a + x +--R a +--R (2) ------------------------- +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 97 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) 2atan(----------------) + asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 98 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 99 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R - \|- x + a + %i x + a x +--R (5) - %i log(-------------------------) + asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 100 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 101 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) %i log(--------------------) - %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 102 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R |- x + a | 2 2 +--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R | 2 2 +--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 103 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 104 14:253 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.254~~~~~$\displaystyle @@ -445,7 +1492,7 @@ $$ <<*>>= )clear all ---S 18 of 28 +--S 105 aa:=integrate(x^3/(a^2-x^2)^(3/2),x) --R --R @@ -457,6 +1504,25 @@ aa:=integrate(x^3/(a^2-x^2)^(3/2),x) --R (x - 2a )\|- x + a - 2a x + 2a --R Type: Union(Expression Integer,...) --E + +--S 106 +bb:=sqrt(a^2-x^2)+a^2/sqrt(a^2-x^2) +--R +--R 2 2 +--R - x + 2a +--R (2) ------------ +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 107 14:254 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R (3) 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.255~~~~~$\displaystyle @@ -468,7 +1534,7 @@ $$ <<*>>= )clear all ---S 19 of 28 +--S 108 aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x) --R --R @@ -483,6 +1549,72 @@ aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x) --R a \|- x + a + a x - a --R Type: Union(Expression Integer,...) --E + +--S 109 +bb:=1/(a^2*sqrt(a^2-x^2))-1/a^3*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 \|- x + a + a +--R - \|- x + a log(----------------) + a +--R x +--R (2) --------------------------------------- +--R +---------+ +--R 3 | 2 2 +--R a \|- x + a +--R Type: Expression Integer +--E + +--S 110 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) + 1 +--R x x +--R (3) ------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 111 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) + 1 +--R (4) ----------------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 112 +ee:=complexNormalize dd +--R +--R x +--R - 2log(-------) + 1 +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 113 14:255 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 2log(\|- 1 ) + 1 +--R (6) ---------------- +--R 3 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.256~~~~~$\displaystyle @@ -493,7 +1625,7 @@ $$ <<*>>= )clear all ---S 20 of 28 +--S 114 aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x) --R --R @@ -506,6 +1638,25 @@ aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x) --R (a x - 2a x)\|- x + a - 2a x + 2a x --R Type: Union(Expression Integer,...) --E + +--S 115 +bb:=-sqrt(a^2-x^2)/(a^4*x)+x/(a^4*sqrt(a^2-x^2)) +--R +--R 2 2 +--R 2x - a +--R (2) --------------- +--R +---------+ +--R 4 | 2 2 +--R a x\|- x + a +--R Type: Expression Integer +--E + +--S 116 14:256 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.257~~~~~$\displaystyle @@ -518,7 +1669,7 @@ $$ <<*>>= )clear all ---S 21 of 28 +--S 117 aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x) --R --R @@ -542,6 +1693,72 @@ aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x) --R (6a x - 8a x )\|- x + a + 2a x - 10a x + 8a x --R Type: Union(Expression Integer,...) --E + +--S 118 +bb:=-1/(2*a^2*x^2*sqrt(a^2-x^2))+3/(2*a^4*sqrt(a^2-x^2))-3/(2*a^5)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 \|- x + a + a 2 3 +--R - 3x \|- x + a log(----------------) + 3a x - a +--R x +--R (2) --------------------------------------------------- +--R +---------+ +--R 5 2 | 2 2 +--R 2a x \|- x + a +--R Type: Expression Integer +--E + +--S 119 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R 3log(----------------) + 3log(----------------) + 2 +--R x x +--R (3) --------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 120 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R 3log(\|- x + a + a) + 3log(\|- x + a - a) - 6log(x) + 2 +--R (4) ------------------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 121 +ee:=complexNormalize dd +--R +--R x +--R - 3log(-------) + 1 +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------------- +--R 5 +--R a +--R Type: Expression Integer +--E + +--S 122 14:257 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 3log(\|- 1 ) + 1 +--R (6) ---------------- +--R 5 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.258~~~~~$\displaystyle\int{(a^2-x^2)^{3/2}}~dx$} @@ -552,7 +1769,7 @@ $$ <<*>>= )clear all ---S 22 of 28 +--S 123 aa:=integrate((a^2-x^2)^(3/2),x) --R --R @@ -579,6 +1796,128 @@ aa:=integrate((a^2-x^2)^(3/2),x) --R (32a x - 64a )\|- x + a + 8x - 64a x + 64a --R Type: Union(Expression Integer,...) --E + +--S 124 +bb:=(x*(a^2-x^2)^(3/2))/4+(3*a^2*x*sqrt(a^2-x^2))/8+3/8*a^4*asin(x/a) +--R +--R +---------+ +--R 3 2 | 2 2 4 x +--R (- 2x + 5a x)\|- x + a + 3a asin(-) +--R a +--R (2) --------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 125 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 4 \|- x + a - a 4 x +--R - 6a atan(----------------) - 3a asin(-) +--R x a +--R (3) ---------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 126 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 127 +ee:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 \|- x + a - a +--R - 3%i a log(--------------------) - 6a atan(----------------) +--R a x +--R (5) ------------------------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 128 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 129 +ff:=atanrule ee +--R +--R (7) +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 - \|- x + a + %i x + a +--R - 3%i a log(--------------------) + 3%i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R ------------------------------------------------------------------------ +--R 8 +--R Type: Expression Complex Integer +--E + +--S 130 +gg:=expandLog ff +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 4 |- x + a 4 | 2 2 +--R - 3%i a log(a |--------- - %i x) - 3%i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 4 | 2 2 4 4 +--R 3%i a log(\|- x + a - %i x - a) + 3%i a log(a) + 3%i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 131 +hh:=rootSimp gg +--R +--R (9) +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3%i a log(%i\|x - a + %i x - a) - 3%i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 4 | 2 2 4 4 +--R 3%i a log(%i\|x - a - %i x - a) + 3%i a log(a) + 3%i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 132 14:258 Schaums and Axiom agree +ii:=complexNormalize hh +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.259~~~~~$\displaystyle\int{x(a^2-x^2)^{3/2}}~dx$} @@ -586,7 +1925,7 @@ $$\int{x(a^2-x^2)^{3/2}}=\frac{(a^2-x^2)^{5/2}}{5}$$ <<*>>= )clear all ---S 23 of 28 +--S 133 aa:=integrate(x*(a^2-x^2)^(3/2),x) --R --R @@ -603,6 +1942,27 @@ aa:=integrate(x*(a^2-x^2)^(3/2),x) --R (5x - 60a x + 80a )\|- x + a - 25a x + 100a x - 80a --R Type: Union(Expression Integer,...) --E + +--S 134 +bb:=-(a^2-x^2)^(5/2)/5 +--R +--R +---------+ +--R 4 2 2 4 | 2 2 +--R (- x + 2a x - a )\|- x + a +--R (2) ------------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 135 14:259 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 5 +--R a +--R (3) - -- +--R 5 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.260~~~~~$\displaystyle\int{x^2(a^2-x^2)^{3/2}}~dx$} @@ -614,7 +1974,7 @@ $$ <<*>>= )clear all ---S 24 of 28 +--S 136 aa:=integrate(x^2*(a^2-x^2)^(3/2),x) --R --R @@ -647,6 +2007,126 @@ aa:=integrate(x^2*(a^2-x^2)^(3/2),x) --R - 1536a --R Type: Union(Expression Integer,...) --E + +--S 137 +bb:=-(x*(a^2-x^2)^(5/2))/6+(a^2*x*(a^2-x^2)^(3/2))/24+(a^4*x*sqrt(a^2-x^2))/16+a^6/16*asin(x/a) +--R +--R +---------+ +--R 5 2 3 4 | 2 2 6 x +--R (- 8x + 14a x - 3a x)\|- x + a + 3a asin(-) +--R a +--R (2) ------------------------------------------------ +--R 48 +--R Type: Expression Integer +--E + +--S 138 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 6 \|- x + a - a 6 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 16 +--R Type: Expression Integer +--E + +--S 139 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 140 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 6 - \|- x + a + %i x + a 6 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 16 +--R Type: Expression Complex Integer +--E + +--S 141 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 142 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 6 \| a 6 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 16 +--R Type: Expression Complex Integer +--E + +--S 143 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 6 |- x + a 6 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 6 | 2 2 6 6 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 16 +--R Type: Expression Complex Integer +--E + +--S 144 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 6 | 2 2 6 6 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 16 +--R Type: Expression Complex Integer +--E + +--S 145 14:260 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.261~~~~~$\displaystyle\int{x^3(a^2-x^2)^{3/2}}~dx$} @@ -656,7 +2136,7 @@ $$ <<*>>= )clear all ---S 25 of 28 +--S 146 aa:=integrate(x^3*(a^2-x^2)^(3/2),x) --R --R @@ -676,6 +2156,27 @@ aa:=integrate(x^3*(a^2-x^2)^(3/2),x) --R - 3920a x + 2240a --R Type: Union(Expression Integer,...) --E + +--S 147 +bb:=(a^2-x^2)^(7/2)/7-(a^2*(a^2-x^2)^(5/2))/5 +--R +--R +---------+ +--R 6 2 4 4 2 6 | 2 2 +--R (- 5x + 8a x - a x - 2a )\|- x + a +--R (2) ---------------------------------------- +--R 35 +--R Type: Expression Integer +--E + +--S 148 14:261 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 7 +--R 2a +--R (3) - --- +--R 35 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.262~~~~~$\displaystyle @@ -687,7 +2188,7 @@ $$ <<*>>= )clear all ---S 26 of 28 +--S 149 aa:=integrate((a^2-x^2)^(3/2)/x,x) --R --R @@ -707,6 +2208,66 @@ aa:=integrate((a^2-x^2)^(3/2)/x,x) --R (3x - 12a )\|- x + a - 9a x + 12a --R Type: Union(Expression Integer,...) --E + +--S 150 +bb:=(a^2-x^2)^(3/2)/3+a^2*sqrt(a^2-x^2)-a^3*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 3 \|- x + a + a 2 2 | 2 2 +--R - 3a log(----------------) + (- x + 4a )\|- x + a +--R x +--R (2) ----------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 151 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R 3 \|- x + a + a 3 \|- x + a - a 3 +--R 3a log(----------------) + 3a log(----------------) + 4a +--R x x +--R (3) --------------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 152 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R 3 | 2 2 3 | 2 2 3 3 +--R 3a log(\|- x + a + a) + 3a log(\|- x + a - a) - 6a log(x) + 4a +--R (4) --------------------------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 153 +ee:=complexNormalize dd +--R +--R 3 x 3 +--R - 6a log(-------) + 4a +--R +----+ +--R | 2 +--R \|- x +--R (5) ----------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 154 14:262 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R 3 +---+ 3 +--R 6a log(\|- 1 ) + 4a +--R (6) -------------------- +--R 3 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.263~~~~~$\displaystyle @@ -718,7 +2279,7 @@ $$ <<*>>= )clear all ---S 27 of 28 +--S 155 aa:=integrate((a^2-x^2)^{3/2}/x^2,x) --R --R @@ -738,6 +2299,127 @@ aa:=integrate((a^2-x^2)^{3/2}/x^2,x) --R (2x - 8a x)\|- x + a - 6a x + 8a x --R Type: Union(Expression Integer,...) --E + +--S 156 +bb:=-(a^2-x^2)^(3/2)/x-(3*x*sqrt(a^2-x^2))/2-3/2*a^2*asin(x/a) +--R +--R +---------+ +--R 2 2 | 2 2 2 x +--R (- x - 2a )\|- x + a - 3a x asin(-) +--R a +--R (2) --------------------------------------- +--R 2x +--R Type: Expression Integer +--E + +--S 157 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R 6a atan(----------------) + 3a asin(-) +--R x a +--R (3) -------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 158 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 159 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 \|- x + a - a +--R 3%i a log(--------------------) + 6a atan(----------------) +--R a x +--R (5) ----------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 160 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 161 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R 3%i a log(--------------------) - 3%i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 162 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R 3%i a log(a |--------- - %i x) + 3%i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R - 3%i a log(\|- x + a - %i x - a) - 3%i a log(a) - 3%i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 163 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R 3%i a log(%i\|x - a + %i x - a) + 3%i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R - 3%i a log(%i\|x - a - %i x - a) - 3%i a log(a) - 3%i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 164 14:263 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.264~~~~~$\displaystyle @@ -749,7 +2431,7 @@ $$ <<*>>= )clear all ---S 28 of 28 +--S 165 aa:=integrate((a^2-x^2)^(3/2)/x^3,x) --R --R @@ -770,6 +2452,63 @@ aa:=integrate((a^2-x^2)^(3/2)/x^3,x) --R Type: Union(Expression Integer,...) --E +--S 166 +bb:=-(a^2-x^2)^(3/2)/(2*x^2)-(3*sqrt(a^2-x^2))/2+3/2*a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a 2 2 | 2 2 +--R 3a x log(----------------) + (- 2x - a )\|- x + a +--R x +--R (2) ----------------------------------------------------- +--R 2 +--R 2x +--R Type: Expression Integer +--E + +--S 167 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R - 3a log(----------------) - 3a log(----------------) - 2a +--R x x +--R (3) ---------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 168 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - 3a log(\|- x + a + a) - 3a log(\|- x + a - a) + 6a log(x) - 2a +--R (4) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 169 +ee:=complexNormalize dd +--R +--R x +--R (5) 3a log(-------) - a +--R +----+ +--R | 2 +--R \|- x +--R Type: Expression Integer +--E + +--S 170 14:264 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R (6) - 3a log(\|- 1 ) - a +--R Type: Expression Integer +--E + )spool )lisp (bye) @ diff --git a/src/input/schaum2.input.pamphlet b/src/input/schaum2.input.pamphlet index 184122e..dfbaf88 100644 --- a/src/input/schaum2.input.pamphlet +++ b/src/input/schaum2.input.pamphlet @@ -742,58 +742,11 @@ cc:=aa-bb --R 15a --R Type: Expression Integer --E -@ -If we had the terms -<<*>>= ---S 52 -t1:=(3*a*x-2*b) ---R ---R (4) 3a x - 2b ---R Type: Polynomial Integer ---E ---S 53 -t2:=(a*x+b) ---R ---R (5) a x + b ---R Type: Polynomial Integer ---E -@ -We can construct the Axiom result -<<*>>= ---S 54 -2*t1*t2*sqrt(t2)/(15*a^2) ---R ---R 2 2 2 +-------+ ---R (6a x + 2a b x - 4b )\|a x + b ---R (6) -------------------------------- ---R 2 ---R 15a ---R Type: Expression Integer ---E -@ -and we can construct the Spiegel result -<<*>>= ---S 55 -2*t1*sqrt(t2^3)/(15*a^2) +--S 52 14:90 Schaums and Axiom agree +dd:=rootSimp cc --R ---R +----------------------------+ ---R | 3 3 2 2 2 3 ---R (6a x - 4b)\|a x + 3a b x + 3a b x + b ---R (7) ------------------------------------------ ---R 2 ---R 15a ---R Type: Expression Integer ---E -@ -the difference of these two depends on -<<*>>= ---S 56 14:90 Axiom cannot simplify this expression -t2*sqrt(t2)-sqrt(t2^3) ---R ---R +----------------------------+ ---R | 3 3 2 2 2 3 +-------+ ---R (8) - \|a x + 3a b x + 3a b x + b + (a x + b)\|a x + b +--R (4) 0 --R Type: Expression Integer --E @ @@ -807,7 +760,7 @@ Note: the sqrt term is almost certainly $\sqrt{(ax+b)}$ <<*>>= )clear all ---S 57 +--S 53 aa:=integrate(x^2*sqrt(a*x+b),x) --R --R @@ -819,27 +772,26 @@ aa:=integrate(x^2*sqrt(a*x+b),x) --R Type: Union(Expression Integer,...) --E ---S 58 -bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^2)*sqrt((a*x+b)^3) +--S 54 +bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^3)*sqrt((a+b*x)^3) --R --R --R +----------------------------+ ---R 2 2 2 | 3 3 2 2 2 3 ---R (30a x - 24a b x + 16b )\|a x + 3a b x + 3a b x + b +--R 2 2 2 | 3 3 2 2 2 3 +--R (30a x - 24a b x + 16b )\|b x + 3a b x + 3a b x + a --R (2) -------------------------------------------------------- ---R 2 +--R 3 --R 105a --R Type: Expression Integer --E ---S 59 14:91 Axiom cannot simplify this expression +--S 55 14:91 Axiom cannot simplify this expression. Schaums typo? cc:=aa-bb ---R --R --R (3) ---R +----------------------------+ ---R 3 2 2 2 | 3 3 2 2 2 3 ---R (- 30a x + 24a b x - 16a b )\|a x + 3a b x + 3a b x + b +--R +----------------------------+ +--R 2 2 2 | 3 3 2 2 2 3 +--R (- 30a x + 24a b x - 16b )\|b x + 3a b x + 3a b x + a --R + --R 3 3 2 2 2 3 +-------+ --R (30a x + 6a b x - 8a b x + 16b )\|a x + b @@ -848,6 +800,20 @@ cc:=aa-bb --R 105a --R Type: Expression Integer --E + +@ +Notice that if we factor the numerator of 'aa' we get an expression that +differs from schaums on by the order of the variables in the square root. +(We can square the term (a*x+b) and drag it under the square root to get +the cubic term). It appears that Schaums has a typo. +<<*>>= +--S 56 +factor numer aa +--R +--R 2 2 2 +-------+ +--R (4) 2(a x + b)(15a x - 12a b x + 8b )\|a x + b +--RType: Factored SparseMultivariatePolynomial(Integer,Kernel Expression Integer) +--E @ \section{\cite{1}:14.92~~~~~$\displaystyle @@ -858,7 +824,7 @@ $$ <<*>>= )clear all ---S 60 +--S 57 aa:=integrate(sqrt(a*x+b)/x,x) --R --R @@ -875,7 +841,7 @@ aa:=integrate(sqrt(a*x+b)/x,x) --R Type: Union(List Expression Integer,...) --E ---S 61 +--S 58 dd:=integrate(1/(x*sqrt(a*x+b)),x) --R --R @@ -889,7 +855,7 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x) --R Type: Union(List Expression Integer,...) --E ---S 62 +--S 59 bb1:=2*sqrt(a*x+b)+b*dd.1 --R --R @@ -903,7 +869,7 @@ bb1:=2*sqrt(a*x+b)+b*dd.1 --R Type: Expression Integer --E ---S 63 +--S 60 bb2:=2*sqrt(a*x+b)+b*dd.2 --R --R @@ -917,7 +883,7 @@ bb2:=2*sqrt(a*x+b)+b*dd.2 --R Type: Expression Integer --E ---S 64 +--S 61 cc11:=bb1-aa.1 --R --R @@ -932,7 +898,7 @@ cc11:=bb1-aa.1 --R Type: Expression Integer --E ---S 65 +--S 62 cc12:=bb1-aa.2 --R --R @@ -947,7 +913,7 @@ cc12:=bb1-aa.2 --R Type: Expression Integer --E ---S 66 +--S 63 cc21:=bb2-aa.1 --R --R @@ -962,7 +928,7 @@ cc21:=bb2-aa.1 --R Type: Expression Integer --E ---S 67 14:92 Axiom cannot simplify this expression +--S 64 14:92 Axiom cannot simplify this expression cc22:=bb2-aa.2 --R --R @@ -986,7 +952,7 @@ $$ <<*>>= )clear all ---S 68 +--S 65 aa:=integrate(sqrt(a*x+b)/x^2,x) --R --R @@ -1008,7 +974,7 @@ aa:=integrate(sqrt(a*x+b)/x^2,x) --R Type: Union(List Expression Integer,...) --E ---S 69 +--S 66 dd:=integrate(1/(x*sqrt(a*x+b)),x) --R --R @@ -1022,7 +988,7 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x) --R Type: Union(List Expression Integer,...) --E ---S 70 +--S 67 bb1:=-sqrt(a*x+b)/x+a/2*dd.1 --R --R @@ -1036,7 +1002,7 @@ bb1:=-sqrt(a*x+b)/x+a/2*dd.1 --R Type: Expression Integer --E ---S 71 +--S 68 bb2:=-sqrt(a*x+b)/x+a/2*dd.2 --R --R @@ -1050,7 +1016,7 @@ bb2:=-sqrt(a*x+b)/x+a/2*dd.2 --R Type: Expression Integer --E ---S 72 +--S 69 cc11:=bb1-aa.1 --R --R @@ -1058,7 +1024,7 @@ cc11:=bb1-aa.1 --R Type: Expression Integer --E ---S 73 +--S 70 cc21:=bb-aa.1 --R --R @@ -1073,7 +1039,7 @@ cc21:=bb-aa.1 --R Type: Expression Integer --E ---S 74 +--S 71 cc12:=bb1-aa.2 --R --R @@ -1088,7 +1054,7 @@ cc12:=bb1-aa.2 --R Type: Expression Integer --E ---S 75 14:93 Schaums and Axiom agree +--S 72 14:93 Schaums and Axiom agree cc22:=bb2-aa.2 --R --R @@ -1106,7 +1072,7 @@ $$ <<*>>= )clear all ---S 76 14:94 Axiom cannot do this integral +--S 73 14:94 Axiom cannot do this integral aa:=integrate(x^m/sqrt(a*x+b),x) --R --R @@ -1128,7 +1094,7 @@ $$ <<*>>= )clear all ---S 77 14:95 Axiom cannot do this integral +--S 74 14:95 Axiom cannot do this integral aa:=integrate(1/(x^m*sqrt(a*x+b)),x) --R --R @@ -1150,7 +1116,7 @@ $$ <<*>>= )clear all ---S 78 14:96 Axiom cannot do this integral +--S 75 14:96 Axiom cannot do this integral aa:=integrate(x^m*sqrt(a*x+b),x) --R --R @@ -1171,7 +1137,7 @@ $$ <<*>>= )clear all ---S 79 14:97 Axiom cannot do this integral +--S 76 14:97 Axiom cannot do this integral aa:=integrate(sqrt(a*x+b)/x^m,x) --R --R @@ -1194,7 +1160,7 @@ Note: 14.98 is the same as 14.97 <<*>>= )clear all ---S 80 14:98 Axiom cannot do this integral +--S 77 14:98 Axiom cannot do this integral aa:=integrate(sqrt(a*x+b)/x^m,x) --R --R @@ -1215,7 +1181,7 @@ $$ <<*>>= )clear all ---S 81 +--S 78 aa:=integrate((a*x+b)^(m/2),x) --R --R @@ -1228,7 +1194,7 @@ aa:=integrate((a*x+b)^(m/2),x) --R Type: Union(Expression Integer,...) --E ---S 82 +--S 79 bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2)) --R --R @@ -1241,7 +1207,7 @@ bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2)) --R Type: Expression Integer --E ---S 83 14:99 Axiom cannot simplify this expression +--S 80 cc:=aa-bb --R --R @@ -1253,6 +1219,33 @@ cc:=aa-bb --R a m + 2a --R Type: Expression Integer --E + +--S 81 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 82 +dd:=explog cc +--R +--R m + 2 m +--R ----- - +--R 2 2 +--R - 2(a x + b) + (2a x + 2b)(a x + b) +--R (5) ----------------------------------------- +--R a m + 2a +--R Type: Expression Integer +--E + +--S 83 14:99 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.100~~~~~$\displaystyle @@ -1292,7 +1285,7 @@ bb:=(2*(a*x+b)^((m+4)/2))/(a^2*(m+4))-(2*b*(a*x+b)^((m+2)/2))/(a^2*(m+2)) --R Type: Expression Integer --E ---S 86 14:100 Axiom cannot simplify this expression +--S 86 cc:=aa-bb --R --R @@ -1311,6 +1304,40 @@ cc:=aa-bb --R a m + 6a m + 8a --R Type: Expression Integer --E + +--S 87 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 88 +dd:=explog cc +--R +--R (5) +--R m + 4 m + 2 +--R ----- ----- +--R 2 2 +--R (- 2m - 4)(a x + b) + (2b m + 8b)(a x + b) +--R + +--R m +--R - +--R 2 2 2 2 2 +--R ((2a m + 4a )x + 2a b m x - 4b )(a x + b) +--R / +--R 2 2 2 2 +--R a m + 6a m + 8a +--R Type: Expression Integer +--E + +--S 89 14:100 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.101~~~~~$\displaystyle @@ -1323,7 +1350,7 @@ $$ <<*>>= )clear all ---S 87 +--S 90 aa:=integrate(x^2*(a*x+b)^(m/2),x) --R --R @@ -1341,7 +1368,7 @@ aa:=integrate(x^2*(a*x+b)^(m/2),x) --R Type: Union(Expression Integer,...) --E ---S 88 +--S 91 bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_ (4*b*(a*x+b)^((m+4)/2))/(a^3*(m+4))+_ (2*b^2*(a*x+b)^((m+2)/2))/(a^3*(m+2)) @@ -1363,7 +1390,7 @@ bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_ --R Type: Expression Integer --E ---S 89 14:101 Axiom cannot simplify this expression +--S 92 cc:=aa-bb --R --R @@ -1390,6 +1417,48 @@ cc:=aa-bb --R a m + 12a m + 44a m + 48a --R Type: Expression Integer --E + +--S 93 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 94 +dd:=explog cc +--R +--R (5) +--R m + 6 m + 4 +--R ----- ----- +--R 2 2 2 2 +--R (- 2m - 12m - 16)(a x + b) + (4b m + 32b m + 48b)(a x + b) +--R + +--R m + 2 +--R ----- +--R 2 2 2 2 2 +--R (- 2b m - 20b m - 48b )(a x + b) +--R + +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((2a m + 12a m + 16a )x + (2a b m + 4a b m)x - 8a b m x + 16b ) +--R * +--R m +--R - +--R 2 +--R (a x + b) +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Expression Integer +--E + +--S 95 14:101 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.102~~~~~$\displaystyle @@ -1401,7 +1470,7 @@ $$ <<*>>= )clear all ---S 90 14:102 Axiom cannot do this integral +--S 96 14:102 Axiom cannot do this integral aa:=integrate((a*x+b)^(m/2)/x,x) --R --R @@ -1423,7 +1492,7 @@ $$ <<*>>= )clear all ---S 91 14:103 Axiom cannot do this integral +--S 97 14:103 Axiom cannot do this integral aa:=integrate((a*x+b)^(m/2)/x^2,x) --R --R @@ -1446,7 +1515,7 @@ $$ <<*>>= )clear all ---S 92 14:104 Axiom cannot do this integral +--S 98 14:104 Axiom cannot do this integral aa:=integrate(1/(x*(a*x+b)^(m/2)),x) --R --R diff --git a/src/input/schaum6.input.pamphlet b/src/input/schaum6.input.pamphlet index 9b6f4f0..854e780 100644 --- a/src/input/schaum6.input.pamphlet +++ b/src/input/schaum6.input.pamphlet @@ -792,7 +792,7 @@ explog:=rule(%e^(n*log(x)) == x^n) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 64 14:140 Schaums and Axiom cannot simplify this expression +--S 64 dd:=explog cc --R --R 2 2 n 2 2 2 2 n - 1 @@ -802,6 +802,13 @@ dd:=explog cc --R (2n - 2)(x + a ) (x + a ) --R Type: Expression Integer --E + +--S 65 14:140 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.141~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^n}}$} @@ -812,7 +819,7 @@ $$ <<*>>= )clear all ---S 65 14:141 Axiom cannot do this integral +--S 66 14:141 Axiom cannot do this integral aa:=integrate(1/(x*(x^2+a^2)^n),x) --R --R @@ -833,7 +840,7 @@ $$ <<*>>= )clear all ---S 66 14:142 Axiom cannot do this integral +--S 67 14:142 Axiom cannot do this integral aa:=integrate(x^m/((x^2+a^2)^n),x) --R --R @@ -854,7 +861,7 @@ $$ <<*>>= )clear all ---S 67 14:143 Axiom cannot do this integral +--S 68 14:143 Axiom cannot do this integral aa:=integrate(1/(x^m*(x^2+a^2)^n),x) --R --R diff --git a/src/input/schaum7.input.pamphlet b/src/input/schaum7.input.pamphlet index ce37f19..fcdd71c 100644 --- a/src/input/schaum7.input.pamphlet +++ b/src/input/schaum7.input.pamphlet @@ -915,7 +915,7 @@ explog:=rule(%e^(n*log(x)) == x^n) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 76 14:159 Axiom cannot simplify this expression +--S 76 dd:=explog cc --R --R 2 2 n 2 2 2 2 n - 1 @@ -926,6 +926,12 @@ dd:=explog cc --R Type: Expression Integer --E +--S 77 14:159 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.160~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)^n}}$} @@ -936,7 +942,7 @@ $$ <<*>>= )clear all ---S 77 14:160 Axiom cannot compute this integral +--S 78 14:160 Axiom cannot compute this integral aa:=integrate(1/(x*(x^2-a^2)^n),x) --R --R @@ -957,7 +963,7 @@ $$ <<*>>= )clear all ---S 78 14:161 Axiom cannot compute this integral +--S 79 14:161 Axiom cannot compute this integral aa:=integrate(x^m/((x^2-a^2)^n),x) --R --R @@ -978,7 +984,7 @@ $$ <<*>>= )clear all ---S 79 14:162 Axiom cannot compute this integral +--S 80 14:162 Axiom cannot compute this integral aa:=integrate(1/(x^m*(x^2-a^2)^n),x) --R --R diff --git a/src/input/schaum8.input.pamphlet b/src/input/schaum8.input.pamphlet index 778d135..10d9ef5 100644 --- a/src/input/schaum8.input.pamphlet +++ b/src/input/schaum8.input.pamphlet @@ -1103,7 +1103,7 @@ explog:=rule(%e^(n*log(x)) == x^n) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 95 14:178 Axiom cannot simplify this expression +--S 95 dd:=explog cc --R --R 2 2 n 2 2 2 2 n - 1 @@ -1114,6 +1114,13 @@ dd:=explog cc --R Type: Expression Integer --E +--S 96 14:178 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.179~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^n}}$} @@ -1124,7 +1131,7 @@ $$ <<*>>= )clear all ---S 96 14:179 Axiom cannot integrate this expression +--S 97 14:179 Axiom cannot integrate this expression aa:=integrate(1/(x*(a^2-x^2)^n),x) --R --R @@ -1145,7 +1152,7 @@ $$ <<*>>= )clear all ---S 97 14:180 Axiom cannot integrate this expression +--S 98 14:180 Axiom cannot integrate this expression aa:=integrate(x^m/((a^2-x^2)^n),x) --R --R @@ -1166,7 +1173,7 @@ $$ <<*>>= )clear all ---S 98 14:181 Axiom cannot integrate this expression +--S 99 14:181 Axiom cannot integrate this expression aa:=integrate(1/(x^m*(a^2-x^2)^n),x) --R --R diff --git a/src/input/schaum9.input.pamphlet b/src/input/schaum9.input.pamphlet index fceb094..d632de7 100644 --- a/src/input/schaum9.input.pamphlet +++ b/src/input/schaum9.input.pamphlet @@ -35,7 +35,7 @@ bb:=log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 3 14:182 Axiom cannot simplify this expression +--S 3 cc:=aa-bb --R --R +-------+ +-------+ @@ -43,6 +43,15 @@ cc:=aa-bb --R (3) - log(\|x + a + x) - log(\|x + a - x) --R Type: Expression Integer --E + +--S 4 14:182 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(a ) +--R Type: Expression Integer +--E + @ This is equal to $-\log(a^2)$ but Axiom cannot prove it. @@ -51,7 +60,7 @@ $$\int{\frac{x}{\sqrt{x^2+a^2}}}=\sqrt{x^2+a^2}$$ <<*>>= )clear all ---S 4 +--S 5 aa:=integrate(x/(sqrt(x^2+a^2)),x) --R --R @@ -65,7 +74,7 @@ aa:=integrate(x/(sqrt(x^2+a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 5 +--S 6 bb:=sqrt(x^2+a^2) --R --R +-------+ @@ -74,7 +83,7 @@ bb:=sqrt(x^2+a^2) --R Type: Expression Integer --E ---S 6 14:183 Schaums and Axiom agree +--S 7 14:183 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -90,7 +99,7 @@ $$ <<*>>= )clear all ---S 7 +--S 8 aa:=integrate(x^2/sqrt(x^2+a^2),x) --R --R @@ -109,7 +118,7 @@ aa:=integrate(x^2/sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 8 +--S 9 bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -120,7 +129,7 @@ bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 9 +--S 10 cc:=aa-bb --R --R +-------+ +-------+ @@ -131,14 +140,14 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 10 +--S 11 logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b)) --R --I (4) c log(b) + c log(a) + %K == c log(a b) + %K --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 11 14:184 Schaums and Axiom differ by a constant +--S 12 14:184 Schaums and Axiom differ by a constant dd:=logmul1 cc --R --R 2 2 @@ -158,7 +167,7 @@ $$ <<*>>= )clear all ---S 12 +--S 13 aa:=integrate(x^3/sqrt(x^2+a^2),x) --R --R @@ -172,7 +181,7 @@ aa:=integrate(x^3/sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 13 +--S 14 bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2) --R --R +-------+ @@ -183,7 +192,7 @@ bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2) --R Type: Expression Integer --E ---S 14 14:185 Schaums and Axiom agree +--S 15 14:185 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -198,7 +207,7 @@ $$ <<*>>= )clear all ---S 15 +--S 16 aa:=integrate(1/(x*sqrt(x^2+a^2)),x) --R --R @@ -210,7 +219,7 @@ aa:=integrate(1/(x*sqrt(x^2+a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 16 +--S 17 bb:=-1/a*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -223,7 +232,7 @@ bb:=-1/a*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 17 +--S 18 cc:=aa-bb --R --R (3) @@ -237,19 +246,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 18 -divlog:=rule(log(a/b) == log(a) - log(b)) ---R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 19 14:186 Axiom cannot simplify this expression -dd:=divlog cc +--S 19 +dd:=expandLog cc --R ---R (5) +--R (4) --R +-------+ +-------+ +-------+ --R | 2 2 | 2 2 | 2 2 --R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) @@ -260,6 +260,15 @@ dd:=divlog cc --R Type: Expression Integer --E +--S 20 14:186 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.187~~~~~$\displaystyle @@ -270,7 +279,7 @@ $$ <<*>>= )clear all ---S 20 +--S 21 aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x) --R --R @@ -282,7 +291,7 @@ aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 21 +--S 22 bb:=-sqrt(x^2+a^2)/(a^2*x) --R --R +-------+ @@ -294,7 +303,7 @@ bb:=-sqrt(x^2+a^2)/(a^2*x) --R Type: Expression Integer --E ---S 22 14:187 Schaums and Axiom differ by a constant +--S 23 14:187 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 1 @@ -314,7 +323,7 @@ $$ <<*>>= )clear all ---S 23 +--S 24 aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x) --R --R @@ -337,7 +346,7 @@ aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 24 +--S 25 bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -351,7 +360,7 @@ bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 25 +--S 26 cc:=aa-bb --R --R (3) @@ -366,19 +375,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 26 -divlog:=rule(log(a/b) == log(a) - log(b)) +--S 27 +dd:=expandLog cc --R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 27 14:188 Axiom cannot simplify this expression -dd:=divlog cc ---R ---R (5) +--R (4) --R +-------+ +-------+ +-------+ --R | 2 2 | 2 2 | 2 2 --R - log(\|x + a + a) + log(\|x + a - x + a) - log(\|x + a - x - a) @@ -389,6 +389,17 @@ dd:=divlog cc --R 2a --R Type: Expression Integer --E + +--S 28 14:188 Schaums and Axiom differ by a constant +ee:=complexNormalize dd + + log(- 1) + (5) -------- + 3 + 2a + Type: Expression Integer +--E + @ \section{\cite{1}:14.189~~~~~$\displaystyle\int{\sqrt{x^2+a^2}}~dx$} @@ -398,7 +409,7 @@ $$ <<*>>= )clear all ---S 28 +--S 29 aa:=integrate(sqrt(x^2+a^2),x) --R --R @@ -417,7 +428,7 @@ aa:=integrate(sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 29 +--S 30 bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -428,7 +439,7 @@ bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 30 14:189 Axiom cannot simplify this expression +--S 31 cc:=aa-bb --R --R +-------+ +-------+ @@ -438,6 +449,17 @@ cc:=aa-bb --R 2 --R Type: Expression Integer --E + +--S 32 14:189 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(a ) +--R (4) - --------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.190~~~~~$\displaystyle\int{x\sqrt{x^2+a^2}}~dx$} @@ -447,7 +469,7 @@ $$ <<*>>= )clear all ---S 31 +--S 33 aa:=integrate(x*sqrt(x^2+a^2),x) --R --R @@ -461,7 +483,7 @@ aa:=integrate(x*sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 32 +--S 34 bb:=(x^2+a^2)^(3/2)/3 --R --R +-------+ @@ -472,7 +494,7 @@ bb:=(x^2+a^2)^(3/2)/3 --R Type: Expression Integer --E ---S 33 14:190 Schaums and Axiom agree +--S 35 14:190 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -489,7 +511,7 @@ $$ <<*>>= )clear all ---S 34 +--S 36 aa:=integrate(x^2*sqrt(x^2+a^2),x) --R --R @@ -508,7 +530,7 @@ aa:=integrate(x^2*sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 35 +--S 37 bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -519,7 +541,7 @@ bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 36 14:191 Axiom cannot simplify this expression +--S 38 cc:=aa-bb --R --R +-------+ +-------+ @@ -529,6 +551,16 @@ cc:=aa-bb --R 8 --R Type: Expression Integer --E + +--S 39 14:191 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R a log(a ) +--R (4) --------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.192~~~~~$\displaystyle @@ -539,7 +571,7 @@ $$ <<*>>= )clear all ---S 37 +--S 40 aa:=integrate(x^3*sqrt(x^2+a^2),x) --R --R @@ -557,7 +589,7 @@ aa:=integrate(x^3*sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 38 +--S 41 bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3 --R --R +-------+ @@ -568,7 +600,7 @@ bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3 --R Type: Expression Integer --E ---S 39 14:192 Schaums and Axiom agree +--S 42 14:192 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -584,7 +616,7 @@ $$ <<*>>= )clear all ---S 40 +--S 43 aa:=integrate(sqrt(x^2+a^2)/x,x) --R --R @@ -603,7 +635,7 @@ aa:=integrate(sqrt(x^2+a^2)/x,x) --R Type: Union(Expression Integer,...) --E ---S 41 +--S 44 bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -614,7 +646,7 @@ bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 42 14:193 Axiom cannot simplify this expression +--S 45 cc:=aa-bb --R --R (3) @@ -629,6 +661,27 @@ cc:=aa-bb --R x --R Type: Expression Integer --E + +--S 46 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R a log(\|x + a + a) - a log(\|x + a - x + a) +--R + +--R +-------+ +--R | 2 2 +--R a log(\|x + a - x - a) - a log(x) +--R Type: Expression Integer +--E + +--S 47 14:193 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R (5) - a log(- 1) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.194~~~~~$\displaystyle @@ -639,7 +692,7 @@ $$ <<*>>= )clear all ---S 43 +--S 48 aa:=integrate(sqrt(x^2+a^2)/x^2,x) --R --R @@ -653,7 +706,7 @@ aa:=integrate(sqrt(x^2+a^2)/x^2,x) --R Type: Union(Expression Integer,...) --E ---S 44 +--S 49 bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -664,7 +717,7 @@ bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 45 14:194 Axiom cannot simplify this expression +--S 50 cc:=aa-bb --R --R +-------+ +-------+ @@ -672,6 +725,15 @@ cc:=aa-bb --R (3) - log(\|x + a + x) - log(\|x + a - x) - 1 --R Type: Expression Integer --E + +--S 51 14:194 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(a ) - 1 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.195~~~~~$\displaystyle @@ -683,7 +745,7 @@ $$ <<*>>= )clear all ---S 46 +--S 52 aa:=integrate(sqrt(x^2+a^2)/x^3,x) --R --R @@ -706,7 +768,7 @@ aa:=integrate(sqrt(x^2+a^2)/x^3,x) --R Type: Union(Expression Integer,...) --E ---S 47 +--S 53 bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -720,7 +782,7 @@ bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 48 +--S 54 cc:=aa-bb --R --R (3) @@ -734,19 +796,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 49 -divlog:=rule(log(a/b) == log(a) - log(b)) ---R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 50 14:195 Axiom cannot simplify this expression -dd:=divlog cc +--S 55 +dd:=expandLog cc --R ---R (5) +--R (4) --R +-------+ +-------+ +-------+ --R | 2 2 | 2 2 | 2 2 --R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) @@ -756,6 +809,15 @@ dd:=divlog cc --R 2a --R Type: Expression Integer --E + +--S 56 14:195 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.196~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^{3/2}}}$} @@ -765,7 +827,7 @@ $$ <<*>>= )clear all ---S 51 +--S 57 aa:=integrate(1/(x^2+a^2)^(3/2),x) --R --R @@ -777,7 +839,7 @@ aa:=integrate(1/(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 52 +--S 58 bb:=x/(a^2*sqrt(x^2+a^2)) --R --R x @@ -788,7 +850,7 @@ bb:=x/(a^2*sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 53 14:196 Schaums and Axiom differ by a constant +--S 59 14:196 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 1 @@ -807,7 +869,7 @@ $$ <<*>>= )clear all ---S 54 +--S 60 aa:=integrate(x/(x^2+a^2)^(3/2),x) --R --R @@ -821,7 +883,7 @@ aa:=integrate(x/(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 55 +--S 61 bb:=-1/sqrt(x^2+a^2) --R --R 1 @@ -832,7 +894,7 @@ bb:=-1/sqrt(x^2+a^2) --R Type: Expression Integer --E ---S 56 14:197 Schaums and Axiom agree +--S 62 14:197 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -849,7 +911,7 @@ $$ <<*>>= )clear all ---S 57 +--S 63 aa:=integrate(x^2/(x^2+a^2)^(3/2),x) --R --R @@ -863,7 +925,7 @@ aa:=integrate(x^2/(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 58 +--S 64 bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -876,7 +938,7 @@ bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 59 14:198 Axiom cannot simplify this expression +--S 65 cc:=aa-bb --R --R +-------+ +-------+ @@ -884,6 +946,14 @@ cc:=aa-bb --R (3) - log(\|x + a + x) - log(\|x + a - x) - 1 --R Type: Expression Integer --E + +--S 66 14:198 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(a ) - 1 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.199~~~~~$\displaystyle @@ -894,7 +964,7 @@ $$ <<*>>= )clear all ---S 60 +--S 67 aa:=integrate(x^3/(x^2+a^2)^(3/2),x) --R --R @@ -908,7 +978,7 @@ aa:=integrate(x^3/(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 61 +--S 68 bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2) --R --R 2 2 @@ -920,7 +990,7 @@ bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2) --R Type: Expression Integer --E ---S 62 14:199 Schaums and Axiom agree +--S 69 14:199 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -937,7 +1007,7 @@ $$ <<*>>= )clear all ---S 63 +--S 70 aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x) --R --R @@ -956,7 +1026,7 @@ aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 64 +--S 71 bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -971,7 +1041,7 @@ bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 65 +--S 72 cc:=aa-bb --R --R (3) @@ -986,19 +1056,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 66 -divlog:=rule(log(a/b) == log(a) - log(b)) ---R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 67 14:200 Axiom cannot simplify this expression -dd:=divlog cc +--S 73 +dd:=expandLog cc --R ---R (5) +--R (4) --R +-------+ +-------+ +-------+ --R | 2 2 | 2 2 | 2 2 --R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) @@ -1009,6 +1070,17 @@ dd:=divlog cc --R a --R Type: Expression Integer --E + +--S 74 14:200 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R 3 +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.201~~~~~$\displaystyle @@ -1019,7 +1091,7 @@ $$ <<*>>= )clear all ---S 68 +--S 75 aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x) --R --R @@ -1031,7 +1103,7 @@ aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 69 +--S 76 bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2)) --R --R 2 2 @@ -1043,7 +1115,7 @@ bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 70 14:201 Schaums and Axiom differ by a constant +--S 77 14:201 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 2 @@ -1064,7 +1136,7 @@ $$ <<*>>= )clear all ---S 71 +--S 78 aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x) --R --R @@ -1091,7 +1163,7 @@ aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 72 +--S 79 bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -1106,7 +1178,7 @@ bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x --R Type: Expression Integer --E ---S 73 +--S 80 cc:=aa-bb --R --R (3) @@ -1121,19 +1193,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 74 -divlog:=rule(log(a/b) == log(a) - log(b)) ---R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 75 14:202 Axiom cannot simplify this expression -dd:=divlog cc +--S 81 +dd:=expandLog cc --R ---R (5) +--R (4) --R +-------+ +-------+ --R | 2 2 | 2 2 --R - 3log(\|x + a + a) + 3log(\|x + a - x + a) @@ -1146,6 +1209,16 @@ dd:=divlog cc --R 2a --R Type: Expression Integer --E + +--S 82 14:202 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3log(- 1) +--R (5) --------- +--R 5 +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.203~~~~~$\displaystyle\int{(x^2+a^2)^{3/2}}~dx$} @@ -1156,7 +1229,7 @@ $$ <<*>>= )clear all ---S 76 +--S 83 aa:=integrate((x^2+a^2)^(3/2),x) --R --R (1) @@ -1177,7 +1250,7 @@ aa:=integrate((x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 77 +--S 84 bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -1188,7 +1261,7 @@ bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 78 14:203 Axiom cannot simplify this expression +--S 85 cc:=aa-bb --R --R +-------+ +-------+ @@ -1198,6 +1271,16 @@ cc:=aa-bb --R 8 --R Type: Expression Integer --E + +--S 86 14:203 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R 3a log(a ) +--R (4) - ---------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.204~~~~~$\displaystyle\int{x(x^2+a^2)^{3/2}}~dx$} @@ -1205,7 +1288,7 @@ $$\int{x(x^2+a^2)^{3/2}}=\frac{(x^2+a^2)^{5/2}}{5}$$ <<*>>= )clear all ---S 79 +--S 87 aa:=integrate(x*(x^2+a^2)^(3/2),x) --R --R @@ -1223,7 +1306,7 @@ aa:=integrate(x*(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 80 +--S 88 bb:=(x^2+a^2)^(5/2)/5 --R --R +-------+ @@ -1234,7 +1317,7 @@ bb:=(x^2+a^2)^(5/2)/5 --R Type: Expression Integer --E ---S 81 14:204 Schaums and Axiom agree +--S 89 14:204 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -1251,7 +1334,7 @@ $$ <<*>>= )clear all ---S 82 +--S 90 aa:=integrate(x^2*(x^2+a^2)^(3/2),x) --R --R @@ -1280,7 +1363,7 @@ aa:=integrate(x^2*(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 83 +--S 91 bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^6/16*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -1291,7 +1374,7 @@ bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^ --R Type: Expression Integer --E ---S 84 14:205 Axiom cannot simplify this expression +--S 92 cc:=aa-bb --R --R +-------+ +-------+ @@ -1301,6 +1384,16 @@ cc:=aa-bb --R 16 --R Type: Expression Integer --E + +--S 93 14:205 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 6 2 +--R a log(a ) +--R (4) --------- +--R 16 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.206~~~~~$\displaystyle\int{x^3(x^2+a^2)^{3/2}}~dx$} @@ -1310,7 +1403,7 @@ $$ <<*>>= )clear all ---S 85 +--S 94 aa:=integrate(x^3*(x^2+a^2)^(3/2),x) --R --R @@ -1340,7 +1433,7 @@ aa:=integrate(x^3*(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 86 +--S 95 bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5 --R --R +-------+ @@ -1351,7 +1444,7 @@ bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5 --R Type: Expression Integer --E ---S 87 14:206 Schaums and Axiom agree +--S 96 14:206 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -1368,7 +1461,7 @@ $$ <<*>>= )clear all ---S 88 +--S 97 aa:=integrate((x^2+a^2)^(3/2)/x,x) --R --R @@ -1391,7 +1484,7 @@ aa:=integrate((x^2+a^2)^(3/2)/x,x) --R Type: Union(Expression Integer,...) --E ---S 89 +--S 98 bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -1404,7 +1497,7 @@ bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 90 +--S 99 cc:=aa-bb --R --R (3) @@ -1420,19 +1513,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 91 -divlog:=rule(log(a/b) == log(a) - log(b)) +--S 100 +dd:=expandLog cc --R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 92 14:207 Axiom cannot simplify this expression -dd:=divlog cc ---R ---R (5) +--R (4) --R +-------+ +-------+ --R 3 | 2 2 3 | 2 2 --R a log(\|x + a + a) - a log(\|x + a - x + a) @@ -1442,6 +1526,14 @@ dd:=divlog cc --R a log(\|x + a - x - a) - a log(x) --R Type: Expression Integer --E + +--S 101 14:207 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3 +--R (5) - a log(- 1) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.208~~~~~$\displaystyle @@ -1453,7 +1545,7 @@ $$ <<*>>= )clear all ---S 93 +--S 102 aa:=integrate((x^2+a^2)^{3/2}/x^2,x) --R --R @@ -1472,7 +1564,7 @@ aa:=integrate((x^2+a^2)^{3/2}/x^2,x) --R Type: Union(Expression Integer,...) --E ---S 94 +--S 103 bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -1483,7 +1575,7 @@ bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 95 14:208 Axiom cannot simplify this expression +--S 104 cc:=aa-bb --R --R +-------+ +-------+ @@ -1493,6 +1585,17 @@ cc:=aa-bb --R 2 --R Type: Expression Integer --E + +--S 105 14:208 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 2 +--R - 3a log(a ) - 2a +--R (4) ------------------ +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.209~~~~~$\displaystyle @@ -1504,7 +1607,7 @@ $$ <<*>>= )clear all ---S 96 +--S 106 aa:=integrate((x^2+a^2)^(3/2)/x^3,x) --R --R @@ -1527,48 +1630,60 @@ aa:=integrate((x^2+a^2)^(3/2)/x^3,x) --R Type: Union(Expression Integer,...) --E ---S 97 -bb:=-(x^2+a^2)^(3/2)/(2*x^3)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x) +--S 107 +bb:=-(x^2+a^2)^(3/2)/(2*x^2)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ ---R | 2 2 +-------+ ---R 3 \|x + a + a 3 2 2 | 2 2 ---R - 3a x log(--------------) + (3x - x - a )\|x + a +--R | 2 2 +-------+ +--R 2 \|x + a + a 2 2 | 2 2 +--R - 3a x log(--------------) + (2x - a )\|x + a --R x ---R (2) ------------------------------------------------------ ---R 3 ---R 2x +--R (2) ------------------------------------------------- +--R 2 +--R 2x --R Type: Expression Integer --E ---S 98 14:209 Axiom cannot simplify this expression +--S 108 cc:=aa-bb --R --R (3) ---R +-------+ +-------+ ---R 5 3 3 | 2 2 6 3 4 | 2 2 ---R ((- 12a x - 3a x )\|x + a + 12a x + 9a x )log(\|x + a - x + a) ---R + ---R +-------+ +-------+ ---R 5 3 3 | 2 2 6 3 4 | 2 2 ---R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(\|x + a - x - a) ---R + ---R +-------+ ---R +-------+ | 2 2 ---R 5 3 3 | 2 2 6 3 4 \|x + a + a ---R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(--------------) ---R x +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 3a log(\|x + a - x + a) + 3a log(\|x + a - x - a) --R + ---R +-------+ ---R 6 5 2 4 2 3 4 2 4 | 2 2 7 6 2 5 ---R (4x - 4x + 7a x - 7a x + 3a x - 3a x)\|x + a - 4x + 4x - 9a x +--R +-------+ +--R | 2 2 +--R \|x + a + a +--R 3a log(--------------) +--R x +--R / +--R 2 +--R Type: Expression Integer +--E + +--S 109 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 3a log(\|x + a + a) - 3a log(\|x + a - x + a) --R + ---R 2 4 4 3 4 2 6 6 ---R 9a x - 6a x + 6a x - a x + a +--R +-------+ +--R | 2 2 +--R 3a log(\|x + a - x - a) - 3a log(x) --R / ---R +-------+ ---R 5 2 3 | 2 2 6 2 4 ---R (8x + 2a x )\|x + a - 8x - 6a x +--R 2 +--R Type: Expression Integer +--E + +--S 110 14:209 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3a log(- 1) +--R (5) - ----------- +--R 2 --R Type: Expression Integer --E