diff --git a/changelog b/changelog index 3956a7b..c94aa65 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20080426 tpd src/input/schaum28.input show Schaums-Axiom equivalence +20080426 tpd src/input/schaum27.input show Schaums-Axiom equivalence +20080426 tpd src/input/schaum26.input show Schaums-Axiom equivalence 20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence 20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence 20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence diff --git a/src/input/schaum26.input.pamphlet b/src/input/schaum26.input.pamphlet index 7bf7244..9bdeec5 100644 --- a/src/input/schaum26.input.pamphlet +++ b/src/input/schaum26.input.pamphlet @@ -18,13 +18,27 @@ $$ )set message auto off )clear all ---S 1 of 16 +--S 1 aa:=integrate(log(x),x) --R --R --R (1) x log(x) - x --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=x*log(x)-x +--R +--R (2) x log(x) - x +--R Type: Expression Integer +--E + +--S 3 14:525 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.526~~~~~$\displaystyle @@ -35,7 +49,7 @@ $$ <<*>>= )clear all ---S 2 of 16 +--S 4 aa:=integrate(x*log(x),x) --R --R @@ -44,7 +58,24 @@ aa:=integrate(x*log(x),x) --R (1) -------------- --R 4 --R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=x^2/2*(log(x)-1/2) +--R +--R 2 2 +--R 2x log(x) - x +--R (2) -------------- +--R 4 +--R Type: Expression Integer --E + +--S 6 14:526 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.527~~~~~$\displaystyle @@ -55,7 +86,7 @@ $$ <<*>>= )clear all ---S 3 of 16 +--S 7 aa:=integrate(x^m*log(x),x) --R --R @@ -65,7 +96,55 @@ aa:=integrate(x^m*log(x),x) --R 2 --R m + 2m + 1 --R Type: Union(Expression Integer,...) ---E +--E + +--S 8 +bb:=x^(m+1)/(m+1)*(log(x)-1/(m+1)) +--R +--R m + 1 +--R ((m + 1)log(x) - 1)x +--R (2) ------------------------- +--R 2 +--R m + 2m + 1 +--R Type: Expression Integer +--E + +--S 9 +cc:=aa-bb +--R +--R m log(x) m + 1 +--R ((m + 1)x log(x) - x)%e + ((- m - 1)log(x) + 1)x +--R (3) ------------------------------------------------------------- +--R 2 +--R m + 2m + 1 +--R Type: Expression Integer +--E + +--S 10 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 11 +dd:=explog cc +--R +--R m + 1 m +--R ((- m - 1)log(x) + 1)x + ((m + 1)x log(x) - x)x +--R (5) ----------------------------------------------------- +--R 2 +--R m + 2m + 1 +--R Type: Expression Integer +--E + +--S 12 14:527 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.528~~~~~$\displaystyle @@ -76,7 +155,7 @@ $$ <<*>>= )clear all ---S 4 of 16 +--S 13 aa:=integrate(log(x)/x,x) --R --R @@ -85,7 +164,24 @@ aa:=integrate(log(x)/x,x) --R (1) ------- --R 2 --R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=1/2*log(x)^2 +--R +--R 2 +--R log(x) +--R (2) ------- +--R 2 +--R Type: Expression Integer --E + +--S 15 14:528 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.529~~~~~$\displaystyle @@ -96,7 +192,7 @@ $$ <<*>>= )clear all ---S 5 of 16 +--S 16 aa:=integrate(log(x)/x^2,x) --R --R @@ -105,6 +201,22 @@ aa:=integrate(log(x)/x^2,x) --R x --R Type: Union(Expression Integer,...) --E + +--S 17 +bb:=-log(x)/x-1/x +--R +--R - log(x) - 1 +--R (2) ------------ +--R x +--R Type: Expression Integer +--E + +--S 18 14:529 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.530~~~~~$\displaystyle @@ -115,14 +227,29 @@ $$ <<*>>= )clear all ---S 6 of 16 +--S 19 aa:=integrate(log(x)^2,x) --R --R --R 2 --R (1) x log(x) - 2x log(x) + 2x --R Type: Union(Expression Integer,...) +--E + +--S 20 +bb:=x*log(x)^2-2*x*log(x)+2*x +--R +--R 2 +--R (2) x log(x) - 2x log(x) + 2x +--R Type: Expression Integer --E + +--S 21 14:530 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.531~~~~~$\displaystyle @@ -133,7 +260,7 @@ $$ <<*>>= )clear all ---S 7 of 16 +--S 22 aa:=integrate(log(x)^n/x,x) --R --R @@ -142,7 +269,52 @@ aa:=integrate(log(x)^n/x,x) --R (1) --------------------- --R n + 1 --R Type: Union(Expression Integer,...) +--E + +--S 23 +bb:=log(x)^(n+1)/(n+1) +--R +--R n + 1 +--R log(x) +--R (2) ----------- +--R n + 1 +--R Type: Expression Integer --E + +--S 24 +cc:=aa-bb +--R +--R n log(log(x)) n + 1 +--R log(x)%e - log(x) +--R (3) ----------------------------------- +--R n + 1 +--R Type: Expression Integer +--E + +--S 25 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 26 +dd:=explog cc +--R +--R n + 1 n +--R - log(x) + log(x)log(x) +--R (5) ----------------------------- +--R n + 1 +--R Type: Expression Integer +--E + +--S 27 14:531 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.532~~~~~$\displaystyle @@ -153,13 +325,28 @@ $$ <<*>>= )clear all ---S 8 of 16 +--S 28 aa:=integrate(1/(x*log(x)),x) --R --R --R (1) log(log(x)) --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=log(log(x)) +--R +--R (2) log(log(x)) +--R Type: Expression Integer +--E + +--S 30 14:532 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.533~~~~~$\displaystyle @@ -171,7 +358,7 @@ $$ <<*>>= )clear all ---S 9 of 16 +--S 31 14:533 Schaums and Axiom agree by definition aa:=integrate(1/log(x),x) --R --R @@ -189,7 +376,7 @@ $$ <<*>>= )clear all ---S 10 of 16 +--S 32 14:534 Axiom cannot compute this integral aa:=integrate(x^m/log(x),x) --R --R @@ -209,7 +396,7 @@ $$ <<*>>= )clear all ---S 11 of 16 +--S 33 14:535 Axiom cannot compute this integral aa:=integrate(log(x)^n,x) --R --R @@ -229,7 +416,7 @@ $$ <<*>>= )clear all ---S 12 of 16 +--S 34 14:536 Axiom cannot compute this integral aa:=integrate(x^m*log(x)^n,x) --R --R @@ -249,7 +436,7 @@ $$ <<*>>= )clear all ---S 13 of 16 +--S 35 aa:=integrate(log(x^2+a^2),x) --R --R @@ -258,6 +445,22 @@ aa:=integrate(log(x^2+a^2),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 36 +bb:=x*log(x^2+a^2)-2*x+2*a*atan(x/a) +--R +--R 2 2 x +--R (2) x log(x + a ) + 2a atan(-) - 2x +--R a +--R Type: Expression Integer +--E + +--S 37 14:537 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.538~~~~~$\displaystyle @@ -268,7 +471,7 @@ $$ <<*>>= )clear all ---S 14 of 16 +--S 38 aa:=integrate(log(x^2-a^2),x) --R --R @@ -276,6 +479,31 @@ aa:=integrate(log(x^2-a^2),x) --R (1) x log(x - a ) + a log(x + a) - a log(x - a) - 2x --R Type: Union(Expression Integer,...) --E + +--S 39 +bb:=x*log(x^2-a^2)-2*x+a*log((x+a)/(x-a)) +--R +--R 2 2 x + a +--R (2) x log(x - a ) + a log(-----) - 2x +--R x - a +--R Type: Expression Integer +--E + +--S 40 +cc:=aa-bb +--R +--R x + a +--R (3) a log(x + a) - a log(x - a) - a log(-----) +--R x - a +--R Type: Expression Integer +--E + +--S 41 14:538 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.539~~~~~$\displaystyle @@ -287,7 +515,7 @@ $$ <<*>>= )clear all ---S 15a of 16 +--S 42 aa:=integrate(x^m*log(x^2+a^2),x) --R --R @@ -296,11 +524,11 @@ aa:=integrate(x^m*log(x^2+a^2),x) --I (1) | log(a + %I )%I d%I --R ++ --R Type: Union(Expression Integer,...) ---E +--E )clear all ---S 15b of 16 +--S 43 14:539 Axiom cannot compute this integral aa:=integrate(x^m*log(x^2-a^2),x) --R --R diff --git a/src/input/schaum27.input.pamphlet b/src/input/schaum27.input.pamphlet index bb9f4d2..cc233e4 100644 --- a/src/input/schaum27.input.pamphlet +++ b/src/input/schaum27.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 22 +--S 1 aa:=integrate(sinh(a*x),x) --R --R cosh(a x) @@ -26,6 +26,22 @@ aa:=integrate(sinh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=cosh(a*x)/a +--R +--R cosh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:540 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.541~~~~~$\displaystyle @@ -36,7 +52,7 @@ $$ <<*>>= )clear all ---S 2 of 22 +--S 4 aa:=integrate(x*sinh(a*x),x) --R --R @@ -46,6 +62,23 @@ aa:=integrate(x*sinh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 5 +bb:=(x*cosh(a*x))/a-sinh(a*x)/a^2 +--R +--R - sinh(a x) + a x cosh(a x) +--R (2) --------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:541 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.542~~~~~$\displaystyle @@ -56,7 +89,7 @@ $$ <<*>>= )clear all ---S 3 of 22 +--S 7 aa:=integrate(x^2*sinh(a*x),x) --R --R @@ -67,6 +100,24 @@ aa:=integrate(x^2*sinh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 8 +bb:=(x^2/a+2/a^3)*cosh(a*x)-(2*x)/a^2*sinh(a*x) +--R +--R 2 2 +--R - 2a x sinh(a x) + (a x + 2)cosh(a x) +--R (2) -------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:542 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.543~~~~~$\displaystyle @@ -77,7 +128,7 @@ $$ <<*>>= )clear all ---S 4 of 22 +--S 10 14:543 Axiom cannot compute this integral aa:=integrate(sinh(a*x)/x,x) --R --R @@ -97,7 +148,7 @@ $$ <<*>>= )clear all ---S 5 of 22 +--S 11 14:544 Axiom cannot compute this integral aa:=integrate(sinh(a*x)/x^2,x) --R --R @@ -118,7 +169,7 @@ $$ <<*>>= )clear all ---S 6 of 22 +--S 12 aa:=integrate(1/sinh(a*x),x) --R --R @@ -127,6 +178,31 @@ aa:=integrate(1/sinh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 13 +bb:=1/a*log(tanh(a*x)/2) +--R +--R tanh(a x) +--R log(---------) +--R 2 +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 14 14:545 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R tanh(a x) +--R - log(---------) - log(sinh(a x) + cosh(a x) + 1) +--R 2 +--R + +--R log(sinh(a x) + cosh(a x) - 1) +--R / +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.546~~~~~$\displaystyle @@ -138,7 +214,7 @@ $$ <<*>>= )clear all ---S 7 of 22 +--S 15 14:546 Axiom cannot compute this integral aa:=integrate(x/sinh(a*x),x) --R --R @@ -158,7 +234,7 @@ $$ <<*>>= )clear all ---S 8 of 22 +--S 16 aa:=integrate(sinh(a*x)^2,x) --R --R @@ -167,6 +243,22 @@ aa:=integrate(sinh(a*x)^2,x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 17 +bb:=(sinh(a*x)*cosh(a*x))/(2*a)-x/2 +--R +--R cosh(a x)sinh(a x) - a x +--R (2) ------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 18 14:547 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.548~~~~~$\displaystyle @@ -177,7 +269,7 @@ $$ <<*>>= )clear all ---S 9 of 22 +--S 19 aa:=integrate(x*sinh(a*x)^2,x) --R --R @@ -188,6 +280,87 @@ aa:=integrate(x*sinh(a*x)^2,x) --R 8a --R Type: Union(Expression Integer,...) --E + +--S 20 +bb:=(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)-x^2/4 +--R +--R 2 2 +--R 2a x sinh(2a x) - cosh(2a x) - 2a x +--R (2) ------------------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 21 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - 2a x sinh(2a x) - sinh(a x) + 4a x cosh(a x)sinh(a x) + cosh(2a x) +--R + +--R 2 +--R - cosh(a x) +--R / +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 22 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 23 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R - 4a x sinh(2a x) + 8a x cosh(a x)sinh(a x) + cosh(2a x) - 2cosh(a x) + 1 +--R -------------------------------------------------------------------------- +--R 2 +--R 16a +--R Type: Expression Integer +--E + +--S 24 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +ee:=coshsqrrule dd +--R +--R - x sinh(2a x) + 2x cosh(a x)sinh(a x) +--R (7) -------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 26 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %K sinh(y + x) - %K sinh(y - x) +--I (8) %K cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 14:548 Axiom cannot simplify this expression +ff:=sinhcoshrule ee +--R +--R (9) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.549~~~~~$\displaystyle @@ -198,7 +371,7 @@ $$ <<*>>= )clear all ---S 10 of 22 +--S 28 aa:=integrate(1/sinh(a*x)^2,x) --R --R @@ -208,6 +381,158 @@ aa:=integrate(1/sinh(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=-coth(a*x)/a +--R +--R coth(a x) +--R (2) - --------- +--R a +--R Type: Expression Integer +--E + +--S 30 +cc:=aa-bb +--R +--R (3) +--R 2 +--R coth(a x)sinh(a x) + 2cosh(a x)coth(a x)sinh(a x) +--R + +--R 2 +--R (cosh(a x) - 1)coth(a x) - 2 +--R / +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 31 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 32 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R 4cosh(a x)coth(a x)sinh(a x) + (cosh(2a x) + 2cosh(a x) - 3)coth(a x) - 4 +--R -------------------------------------------------------------------------- +--R 2 +--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a +--R Type: Expression Integer +--E + +--S 33 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 +ee:=coshsqrrule dd +--R +--R 2cosh(a x)coth(a x)sinh(a x) + (cosh(2a x) - 1)coth(a x) - 2 +--R (7) ------------------------------------------------------------ +--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 35 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--I %B sinh(y + x) - %B sinh(y - x) +--I (8) %B cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 36 +ff:=sinhcoshrule ee +--R +--R coth(a x)sinh(2a x) + (cosh(2a x) - 1)coth(a x) - 2 +--R (9) --------------------------------------------------- +--R a sinh(2a x) + a cosh(2a x) - a +--R Type: Expression Integer +--E + +--S 37 +cothrule:=rule(coth(x) == cosh(x)/sinh(x)) +--R +--R cosh(x) +--R (10) coth(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 38 +gg:=cothrule ff +--R +--R cosh(a x)sinh(2a x) - 2sinh(a x) + cosh(a x)cosh(2a x) - cosh(a x) +--R (11) ------------------------------------------------------------------ +--R a sinh(a x)sinh(2a x) + (a cosh(2a x) - a)sinh(a x) +--R Type: Expression Integer +--E + +--S 39 +hh:=sinhcoshrule gg +--R +--R sinh(3a x) - 3sinh(a x) + 2cosh(a x)cosh(2a x) - 2cosh(a x) +--R (12) ----------------------------------------------------------- +--R a sinh(3a x) + 2a sinh(a x)sinh(2a x) - 3a sinh(a x) +--R Type: Expression Integer +--E + +--S 40 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %M cosh(y + x) - %M cosh(y - x) +--I (13) %M sinh(x)sinh(y) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 41 +ii:=sinhsinhrule gg +--R +--R 2cosh(a x)sinh(2a x) - 4sinh(a x) + 2cosh(a x)cosh(2a x) - 2cosh(a x) +--R (14) --------------------------------------------------------------------- +--R (2a cosh(2a x) - 2a)sinh(a x) + a cosh(3a x) - a cosh(a x) +--R Type: Expression Integer +--E + +--S 42 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %N cosh(y + x) + %N cosh(y - x) +--I (15) %N cosh(x)cosh(y) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 43 +jj:=coshcoshrule ii +--R +--R 2cosh(a x)sinh(2a x) - 4sinh(a x) + cosh(3a x) - cosh(a x) +--R (16) ---------------------------------------------------------- +--R (2a cosh(2a x) - 2a)sinh(a x) + a cosh(3a x) - a cosh(a x) +--R Type: Expression Integer +--E + +--S 44 14:549 Schaums and Axiom differ by a constant +kk:=sinhcoshrule jj +--R +--R 1 +--R (17) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.550~~~~~$\displaystyle @@ -218,7 +543,7 @@ $$ <<*>>= )clear all ---S 11 of 22 +--S 45 aa:=integrate(sinh(a*x)*sinh(p*x),x) --R --R @@ -228,6 +553,36 @@ aa:=integrate(sinh(a*x)*sinh(p*x),x) --R (p - a )sinh(a x) + (- p + a )cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 46 +bb:=(sinh(a+p)*x)/(2*(a+p))-(sinh(a-p)*x)/(2*(a-p)) +--R +--R (p - a)x sinh(p + a) + (- p - a)x sinh(p - a) +--R (2) --------------------------------------------- +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 47 14:550 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2a cosh(a x)sinh(p x) +--R + +--R 2 +--R ((- p + a)x sinh(p + a) + (p + a)x sinh(p - a))sinh(a x) +--R + +--R 2 +--R - 2p cosh(p x)sinh(a x) + (p - a)x cosh(a x) sinh(p + a) +--R + +--R 2 +--R (- p - a)x cosh(a x) sinh(p - a) +--R / +--R 2 2 2 2 2 2 +--R (2p - 2a )sinh(a x) + (- 2p + 2a )cosh(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.551~~~~~$\displaystyle @@ -238,7 +593,7 @@ $$ <<*>>= )clear all ---S 12 of 22 +--S 48 aa:=integrate(sinh(a*x)*sin(p*x),x) --R --R @@ -255,6 +610,71 @@ aa:=integrate(sinh(a*x)*sin(p*x),x) --R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 49 +bb:=(a*cosh(a*x)*sin(p*x)-p*sinh(a*x)*cos(p*x))/(a^2+p^2) +--R +--R - p cos(p x)sinh(a x) + a cosh(a x)sin(p x) +--R (2) ------------------------------------------- +--R 2 2 +--R p + a +--R Type: Expression Integer +--E + +--S 50 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (a sin(p x) + p cos(p x))sinh(a x) + (- a cosh(a x) + a)sin(p x) +--R + +--R 2 +--R - p cos(p x)cosh(a x) + p cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 51 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 52 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R (a cosh(2a x) - 2a cosh(a x) + a)sin(p x) + p cos(p x)cosh(2a x) +--R + +--R 2 +--R - 2p cos(p x)cosh(a x) + p cos(p x) +--R / +--R 2 2 2 2 +--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x) +--R Type: Expression Integer +--E + +--S 53 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 54 14:551 Schaums and Axiom agree +ee:=coshsqrrule dd +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.552~~~~~$\displaystyle @@ -265,7 +685,7 @@ $$ <<*>>= )clear all ---S 13 of 22 +--S 55 aa:=integrate(sinh(a*x)*cos(p*x),x) --R --R @@ -282,6 +702,71 @@ aa:=integrate(sinh(a*x)*cos(p*x),x) --R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 56 +bb:=(a*cosh(a*x)*cos(p*x)+p*sinh(a*x)*sin(p*x))/(a^2+p^2) +--R +--R p sin(p x)sinh(a x) + a cos(p x)cosh(a x) +--R (2) ----------------------------------------- +--R 2 2 +--R p + a +--R Type: Expression Integer +--E + +--S 57 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- p sin(p x) + a cos(p x))sinh(a x) + (p cosh(a x) - p)sin(p x) +--R + +--R 2 +--R - a cos(p x)cosh(a x) + a cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 58 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 59 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R (- p cosh(2a x) + 2p cosh(a x) - p)sin(p x) + a cos(p x)cosh(2a x) +--R + +--R 2 +--R - 2a cos(p x)cosh(a x) + a cos(p x) +--R / +--R 2 2 2 2 +--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x) +--R Type: Expression Integer +--E + +--S 60 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 61 14:552 Schaums and Axiom agree +ee:=coshsqrrule dd +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.553~~~~~$\displaystyle @@ -293,7 +778,7 @@ $$ <<*>>= )clear all ---S 14 of 22 +--S 62 aa:=integrate(1/(p+q*sinh(a*x)),x) --R --R @@ -322,6 +807,60 @@ aa:=integrate(1/(p+q*sinh(a*x)),x) --R a\|q + p --R Type: Union(Expression Integer,...) --E + +--S 63 +bb:=1/(a*sqrt(p^2+q^2))*log((q*%e^(a*x)+p-sqrt(p^2+q^2))/(q*%e^(a*x)+p+sqrt(p^2+q^2))) +--R +--R +-------+ +--R | 2 2 a x +--R - \|q + p + q %e + p +--R log(--------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q + p + q %e + p +--R (2) ------------------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 64 14:553 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 a x +--R - \|q + p + q %e + p +--R - log(--------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q + p + q %e + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.554~~~~~$\displaystyle @@ -333,7 +872,7 @@ $$ <<*>>= )clear all ---S 15 of 22 +--S 65 aa:=integrate(1/(p*q*sinh(a*x))^2,x) --R --R @@ -344,6 +883,137 @@ aa:=integrate(1/(p*q*sinh(a*x))^2,x) --R a p q sinh(a x) + 2a p q cosh(a x)sinh(a x) + a p q cosh(a x) - a p q --R Type: Union(Expression Integer,...) --E + +--S 66 +t1:=integrate(1/(p+q*sinh(a*x)),x) +--R +--R (2) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Union(Expression Integer,...) +--E + +--S 67 +bb:=(-q*cosh(a*x))/(a*(p^2+q^2)*(p+q*sinh(a*x)))+p/(p^2+q^2)*t1 +--R +--R (3) +--R 2 +--R (p q sinh(a x) + p ) +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 +--R - q cosh(a x)\|q + p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q + a p q)sinh(a x) + a p q + a p )\|q + p +--R Type: Expression Integer +--E + +--S 68 14:554 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (4) +--R 3 3 3 3 3 4 2 2 +--R - p q sinh(a x) + (- 2p q cosh(a x) - p q )sinh(a x) +--R + +--R 3 3 2 4 2 3 3 4 2 2 +--R (- p q cosh(a x) - 2p q cosh(a x) + p q )sinh(a x) - p q cosh(a x) +--R + +--R 4 2 +--R p q +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R 2 3 2 2 3 2 3 2 +--R p q cosh(a x)sinh(a x) + (2p q cosh(a x) - 2q - 2p q)sinh(a x) +--R + +--R 2 3 3 2 3 2 3 +--R p q cosh(a x) - p q cosh(a x) - 2p q - 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R / +--R 2 5 4 3 3 +--R (a p q + a p q )sinh(a x) +--R + +--R 2 5 4 3 3 4 5 2 2 +--R ((2a p q + 2a p q )cosh(a x) + a p q + a p q )sinh(a x) +--R + +--R 2 5 4 3 2 3 4 5 2 2 5 +--R (a p q + a p q )cosh(a x) + (2a p q + 2a p q )cosh(a x) - a p q +--R + +--R 4 3 +--R - a p q +--R * +--R sinh(a x) +--R + +--R 3 4 5 2 2 3 4 5 2 +--R (a p q + a p q )cosh(a x) - a p q - a p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.555~~~~~$\displaystyle @@ -363,7 +1033,7 @@ $$ <<*>>= )clear all ---S 16 of 22 +--S 69 aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x) --R --R @@ -426,6 +1096,180 @@ aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 70 +bb1:=1/(a*p*sqrt(q^2-p^2))*atan((sqrt(q^2-p^2)*tanh(a*x))/p) +--R +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q - p +--R atan(-------------------) +--R p +--R (2) ------------------------- +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +--S 71 +bb2:=1/(2*a*p*sqrt(p^2-q^2))*log((p+sqrt(p^2-q^2)*tanh(a*x))/(p-sqrt(p^2-q^2)*tanh(a*x))) +--R +--R +---------+ +--R | 2 2 +--R - tanh(a x)\|- q + p - p +--R log(---------------------------) +--R +---------+ +--R | 2 2 +--R tanh(a x)\|- q + p - p +--R (3) -------------------------------- +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R Type: Expression Integer +--E + +--S 72 +cc1:=aa.1-bb1 +--R +--R (4) +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) - 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) - 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (- 2q + 4p )cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 tanh(a x)\|q - p +--R - 2\|- q + p atan(-------------------) +--R p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 73 +cc2:=aa.2-bb1 +--R +--R (5) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q - p +--R - atan(-------------------) +--R p +--R + +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +--S 74 +cc3:=aa.2-bb1 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q - p +--R - atan(-------------------) +--R p +--R + +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +--S 75 14:555 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (7) +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 - tanh(a x)\|- q + p - p +--R - \|q - p log(---------------------------) +--R +---------+ +--R | 2 2 +--R tanh(a x)\|- q + p - p +--R + +--R +---------+ +--R | 2 2 +--R 2\|- q + p +--R * +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.556~~~~~$\displaystyle @@ -437,7 +1281,7 @@ $$ <<*>>= )clear all ---S 17 of 22 +--S 76 aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x) --R --R @@ -500,6 +1344,113 @@ aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 77 +bb:=1/(2*a*p*sqrt(p^2+q^2))*log((p+sqrt(p^2+q^2)*tanh(a*x))/(p-sqrt(p^2+q^2)*tanh(a*x))) +--R +--R +-------+ +--R | 2 2 +--R - tanh(a x)\|q + p - p +--R log(-------------------------) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q + p - p +--R (2) ------------------------------ +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Expression Integer +--E + +--S 78 +cc1:=aa.1-bb +--R +--R (3) +--R +-------+ +--R | 2 2 +--R \|q + p +--R * +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) - 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) - 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (- 2q + 4p )cosh(a x) + q +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 - tanh(a x)\|q + p - p +--R - \|- q + p log(-------------------------) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q + p - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q + p +--R Type: Expression Integer +--E + +--S 79 14:556 Axiom cannot simplify this expression +cc2:=aa.2-bb +--R +--R (4) +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 - tanh(a x)\|q + p - p +--R - \|q - p log(-------------------------) +--R +-------+ +--R | 2 2 +--R tanh(a x)\|q + p - p +--R + +--R +-------+ +--R | 2 2 +--R 2\|q + p +--R * +--R atan +--R 2 2 2 2 2 2 2 +--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|q - p \|q + p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.557~~~~~$\displaystyle @@ -510,7 +1461,7 @@ $$ <<*>>= )clear all ---S 18 of 22 +--S 80 14:557 Axiom cannot compute this integral aa:=integrate(x^m*sinh(a*x),x) --R --R @@ -530,7 +1481,7 @@ $$ <<*>>= )clear all ---S 19 of 22 +--S 81 14:558 Axiom cannot compute this integral aa:=integrate(sinh(a*x)^n,x) --R --R @@ -550,16 +1501,17 @@ $$ <<*>>= )clear all ---S 20 of 22 -aa:=integrate(sinh(a*x)/a^n,x) +--S 82 14:559 Axiom cannot compute this integral +aa:=integrate(sinh(a*x)/x^n,x) --R ---R 2 2 ---R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 ---R (1) ------------------------------------------------- ---R n ---R (2a sinh(a x) + 2a cosh(a x))a +--R x +--I ++ sinh(%T a) +--I (3) | ---------- d%T +--R ++ n +--I %T --R Type: Union(Expression Integer,...) --E + @ \section{\cite{1}:14.560~~~~~$\displaystyle @@ -571,7 +1523,7 @@ $$ <<*>>= )clear all ---S 21 of 22 +--S 83 14:560 Axiom cannot compute this integral aa:=integrate(1/sinh(a*x)^n,x) --R --R @@ -594,7 +1546,7 @@ $$ <<*>>= )clear all ---S 22 of 22 +--S 84 14:561 Axiom cannot compute this integral aa:=integrate(x/sinh(a*x)^n,x) --R --R diff --git a/src/input/schaum28.input.pamphlet b/src/input/schaum28.input.pamphlet index b3fc3c6..5c68b06 100644 --- a/src/input/schaum28.input.pamphlet +++ b/src/input/schaum28.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 28 +--S 1 aa:=integrate(cosh(a*x),x) --R --R @@ -27,6 +27,22 @@ aa:=integrate(cosh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=sinh(a*x)/a +--R +--R sinh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:562 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.563~~~~~$\displaystyle @@ -37,7 +53,7 @@ $$ <<*>>= )clear all ---S 2 of 28 +--S 4 aa:=integrate(x*cosh(a*x),x) --R --R @@ -47,6 +63,23 @@ aa:=integrate(x*cosh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 5 +bb:=(x*sinh(a*x))/a-cosh(a*x)/a^2 +--R +--R a x sinh(a x) - cosh(a x) +--R (2) ------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:563 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.564~~~~~$\displaystyle @@ -57,7 +90,7 @@ $$ <<*>>= )clear all ---S 3 of 28 +--S 7 aa:=integrate(x^2*cosh(a*x),x) --R --R @@ -68,6 +101,24 @@ aa:=integrate(x^2*cosh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 8 +bb:=-(2*x*cosh(a*x))/a^2+(x^2/a+2/a^3)*sinh(a*x) +--R +--R 2 2 +--R (a x + 2)sinh(a x) - 2a x cosh(a x) +--R (2) ------------------------------------ +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:564 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.565~~~~~$\displaystyle @@ -80,7 +131,7 @@ $$ <<*>>= )clear all ---S 4 of 28 +--S 10 14:565 Axiom cannot compute this integral aa:=integrate(cosh(a*x)/x,x) --R --R @@ -100,7 +151,7 @@ $$ <<*>>= )clear all ---S 5 of 28 +--S 11 14:566 Axiom cannot compute this integral aa:=integrate(cosh(a*x)/x^2,x) --R --R @@ -121,7 +172,7 @@ $$ <<*>>= )clear all ---S 6 of 28 +--S 12 aa:=integrate(1/cosh(a*x),x) --R --R @@ -130,6 +181,33 @@ aa:=integrate(1/cosh(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 13 +bb:=2/a*atan(%e^(a*x)) +--R +--R a x +--R 2atan(%e ) +--R (2) ------------ +--R a +--R Type: Expression Integer +--E + +--S 14 +cc:=aa-bb +--R +--R a x +--R 2atan(sinh(a x) + cosh(a x)) - 2atan(%e ) +--R (3) ------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 15 14:567 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.568~~~~~$\displaystyle @@ -141,7 +219,7 @@ $$ <<*>>= )clear all ---S 7 of 28 +--S 16 14:568 Axiom cannot compute this integral aa:=integrate(x/cosh(a*x),x) --R --R @@ -161,7 +239,7 @@ $$ <<*>>= )clear all ---S 8 of 28 +--S 17 aa:=integrate(cosh(a*x)^2,x) --R --R @@ -170,6 +248,42 @@ aa:=integrate(cosh(a*x)^2,x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 18 +bb:=x/2+(sinh(a*x)*cosh(a*x))/2 +--R +--R cosh(a x)sinh(a x) + x +--R (2) ---------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 19 +cc:=aa-bb +--R +--R (- a + 1)cosh(a x)sinh(a x) +--R (3) --------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 20 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %B sinh(y + x) - %B sinh(y - x) +--I (4) %B cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 21 14:569 SCHAUMS AND AXIOM DISAGREE? +dd:=sinhcoshrule cc +--R +--R (- a + 1)sinh(2a x) +--R (5) ------------------- +--R 4a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.570~~~~~$\displaystyle @@ -180,7 +294,7 @@ $$ <<*>>= )clear all ---S 9 of 28 +--S 22 aa:=integrate(x*cosh(a*x)^2,x) --R --R @@ -191,6 +305,87 @@ aa:=integrate(x*cosh(a*x)^2,x) --R 8a --R Type: Union(Expression Integer,...) --E + +--S 23 +bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2) +--R +--R 2 2 +--R 2a x sinh(2a x) - cosh(2a x) + 2a x +--R (2) ------------------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 24 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - 2a x sinh(2a x) - sinh(a x) + 4a x cosh(a x)sinh(a x) + cosh(2a x) +--R + +--R 2 +--R - cosh(a x) +--R / +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 25 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 26 +dd:=sinhsqrrule cc +--R +--R (5) +--R 2 +--R - 4a x sinh(2a x) + 8a x cosh(a x)sinh(a x) + cosh(2a x) - 2cosh(a x) + 1 +--R -------------------------------------------------------------------------- +--R 2 +--R 16a +--R Type: Expression Integer +--E + +--S 27 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 28 +ee:=coshsqrrule dd +--R +--R - x sinh(2a x) + 2x cosh(a x)sinh(a x) +--R (7) -------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 29 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %S sinh(y + x) - %S sinh(y - x) +--I (8) %S cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 30 14:570 Schaums and Axiom agree +ff:=sinhcoshrule ee +--R +--R (9) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.571~~~~~$\displaystyle @@ -201,7 +396,7 @@ $$ <<*>>= )clear all ---S 10 of 28 +--S 31 aa:=integrate(1/cosh(a*x)^2,x) --R --R @@ -211,6 +406,176 @@ aa:=integrate(1/cosh(a*x)^2,x) --R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 32 +bb:=tanh(a*x)/a +--R +--R tanh(a x) +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 33 +cc:=aa-bb +--R +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2 +--R (3) ------------------------------------------------------------------ +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 34 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 +dd:=tanhrule cc +--R +--R (5) +--R 3 2 2 +--R - sinh(a x) - 2cosh(a x)sinh(a x) + (- cosh(a x) - 1)sinh(a x) +--R + +--R - 2cosh(a x) +--R / +--R 2 2 3 +--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x) +--R Type: Expression Integer +--E + +--S 36 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 37 +ee:=sinhsqrrule dd +--R +--R (7) +--R 3 2 +--R - 2sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - 2cosh(a x)cosh(2a x) +--R + +--R - 2cosh(a x) +--R / +--R 2 3 +--R 4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x) +--R + +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 38 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (8) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 39 +ff:=coshsqrrule ee +--R +--R (9) +--R 3 +--R - 2sinh(a x) + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x) +--R + +--R - 2cosh(a x) +--R / +--R 3 +--R (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x) +--R + +--R a cosh(a x) +--R Type: Expression Integer +--E + +--S 40 +sinhcuberule:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x)) +--R +--R 3 sinh(3x) - 3sinh(x) +--R (10) sinh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 41 +gg:=sinhcuberule ff +--R +--R (11) +--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x) +--R + +--R - 4cosh(a x) +--R / +--R 3 +--R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(a x)cosh(2a x) + 4a cosh(a x) +--R + +--R 2a cosh(a x) +--R Type: Expression Integer +--E + +--S 42 +coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x)) +--R +--R 3 cosh(3x) - 3cosh(x) +--R (12) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 43 +hh:=coshcuberule gg +--R +--R (13) +--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x) +--R + +--R - 4cosh(a x) +--R / +--R (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) + 2a cosh(a x)cosh(2a x) +--R + +--R - a cosh(a x) +--R Type: Expression Integer +--E + +--S 44 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %U cosh(y + x) + %U cosh(y - x) +--I (14) %U cosh(x)cosh(y) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 45 +ii:=coshcoshrule hh +--R +--R (15) +--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 2cosh(3a x) - 6cosh(a x) +--R ---------------------------------------------------------------------- +--R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(3a x) +--R Type: Expression Integer +--E + +--S 46 14:571 SCHAUMS AND AXIOM DISAGREE? +jj:=complexNormalize ii +--R +--R a x 2 +--R - 2(%e ) - 2 +--R (16) --------------- +--R a x 2 +--R 2a (%e ) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.572~~~~~$\displaystyle @@ -221,7 +586,7 @@ $$ <<*>>= )clear all ---S 11 of 28 +--S 47 aa:=integrate(cosh(a*x)*cosh(p*x),x) --R --R @@ -231,6 +596,106 @@ aa:=integrate(cosh(a*x)*cosh(p*x),x) --R (p - a )sinh(a x) + (- p + a )cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 48 +bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p)) +--R +--R (p - a)x sinh(p + a) + (p + a)x sinh(p - a) +--R (2) ------------------------------------------- +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 49 +cc:=aa-bb +--R +--R (3) +--R - 2p cosh(a x)sinh(p x) +--R + +--R 2 +--R ((- p + a)x sinh(p + a) + (- p - a)x sinh(p - a))sinh(a x) +--R + +--R 2 +--R 2a cosh(p x)sinh(a x) + (p - a)x cosh(a x) sinh(p + a) +--R + +--R 2 +--R (p + a)x cosh(a x) sinh(p - a) +--R / +--R 2 2 2 2 2 2 +--R (2p - 2a )sinh(a x) + (- 2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 50 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (4) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 51 +dd:=sinhsqrrule cc +--R +--R (5) +--R - 4p cosh(a x)sinh(p x) + 4a cosh(p x)sinh(a x) +--R + +--R 2 +--R ((- p + a)x cosh(2a x) + (2p - 2a)x cosh(a x) + (p - a)x)sinh(p + a) +--R + +--R 2 +--R ((- p - a)x cosh(2a x) + (2p + 2a)x cosh(a x) + (p + a)x)sinh(p - a) +--R / +--R 2 2 2 2 2 2 2 +--R (2p - 2a )cosh(2a x) + (- 4p + 4a )cosh(a x) - 2p + 2a +--R Type: Expression Integer +--E + +--S 52 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (6) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 53 +ee:=coshsqrrule dd +--R +--R (7) +--R 2p cosh(a x)sinh(p x) - 2a cosh(p x)sinh(a x) + (- p + a)x sinh(p + a) +--R + +--R (- p - a)x sinh(p - a) +--R / +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 54 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %V sinh(y + x) - %V sinh(y - x) +--I (8) %V cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 55 14:572 Schaums and Axiom agree +ff:=sinhcoshrule ee +--R +--R (9) +--R (p - a)sinh((p + a)x) + (p + a)sinh((p - a)x) + (- p + a)x sinh(p + a) +--R + +--R (- p - a)x sinh(p - a) +--R / +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.573~~~~~$\displaystyle @@ -241,7 +706,7 @@ $$ <<*>>= )clear all ---S 12 of 28 +--S 56 aa:=integrate(cosh(a*x)*sin(p*x),x) --R --R @@ -258,6 +723,70 @@ aa:=integrate(cosh(a*x)*sin(p*x),x) --R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 57 +bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2) +--R +--R a sin(p x)sinh(a x) - p cos(p x)cosh(a x) +--R (2) ----------------------------------------- +--R 2 2 +--R p + a +--R Type: Expression Integer +--E + +--S 58 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (- a sin(p x) - p cos(p x))sinh(a x) + (a cosh(a x) - a)sin(p x) +--R + +--R 2 +--R p cos(p x)cosh(a x) - p cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 59 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (4) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 60 +dd:=coshsqrrule cc +--R +--R (5) +--R 2 +--R (- 2a sin(p x) - 2p cos(p x))sinh(a x) + (a cosh(2a x) - a)sin(p x) +--R + +--R p cos(p x)cosh(2a x) - p cos(p x) +--R / +--R 2 2 2 2 +--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x) +--R Type: Expression Integer +--E + +--S 61 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 62 14:573 Schaums and Axiom agree +ee:=sinhsqrrule dd +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.574~~~~~$\displaystyle @@ -268,7 +797,7 @@ $$ <<*>>= )clear all ---S 13 of 28 +--S 63 aa:=integrate(cosh(a*x)*cos(p*x),x) --R --R @@ -285,6 +814,70 @@ aa:=integrate(cosh(a*x)*cos(p*x),x) --R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) --R Type: Union(Expression Integer,...) --E + +--S 64 +bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2) +--R +--R a cos(p x)sinh(a x) + p cosh(a x)sin(p x) +--R (2) ----------------------------------------- +--R 2 2 +--R p + a +--R Type: Expression Integer +--E + +--S 65 +cc:=aa-bb +--R +--R (3) +--R 2 2 +--R (p sin(p x) - a cos(p x))sinh(a x) + (- p cosh(a x) + p)sin(p x) +--R + +--R 2 +--R a cos(p x)cosh(a x) - a cos(p x) +--R / +--R 2 2 2 2 +--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x) +--R Type: Expression Integer +--E + +--S 66 +coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--R +--R 2 cosh(2x) + 1 +--R (4) cosh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 67 +dd:=coshsqrrule cc +--R +--R (5) +--R 2 +--R (2p sin(p x) - 2a cos(p x))sinh(a x) + (- p cosh(2a x) + p)sin(p x) +--R + +--R a cos(p x)cosh(2a x) - a cos(p x) +--R / +--R 2 2 2 2 +--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x) +--R Type: Expression Integer +--E + +--S 68 +sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (6) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 69 14:574 Schaums and Axiom agree +ee:=sinhsqrrule dd +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.575~~~~~$\displaystyle @@ -295,7 +888,7 @@ $$ <<*>>= )clear all ---S 14 of 28 +--S 70 aa:=integrate(1/(cosh(a*x)+1),x) --R --R @@ -304,6 +897,112 @@ aa:=integrate(1/(cosh(a*x)+1),x) --R a sinh(a x) + a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 71 +bb:=1/a*tanh((a*x)/2) +--R +--R a x +--R tanh(---) +--R 2 +--R (2) --------- +--R a +--R Type: Expression Integer +--E + +--S 72 +cc:=aa-bb +--R +--R a x +--R (- sinh(a x) - cosh(a x) - 1)tanh(---) - 2 +--R 2 +--R (3) ------------------------------------------ +--R a sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 73 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 74 +dd:=tanhrule cc +--R +--R a x a x a x +--R - sinh(---)sinh(a x) + (- cosh(a x) - 1)sinh(---) - 2cosh(---) +--R 2 2 2 +--R (5) -------------------------------------------------------------- +--R a x a x a x +--R a cosh(---)sinh(a x) + a cosh(---)cosh(a x) + a cosh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 75 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %B sinh(y + x) - %B sinh(y - x) +--I (6) %B cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 76 +ee:=sinhcoshrule dd +--R +--R 3a x a x a x a x +--R - sinh(----) - 2sinh(---)sinh(a x) - sinh(---) - 4cosh(---) +--R 2 2 2 2 +--R (7) ----------------------------------------------------------------- +--R 3a x a x a x a x +--R a sinh(----) + a sinh(---) + 2a cosh(---)cosh(a x) + 2a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 77 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %B cosh(y + x) - %B cosh(y - x) +--I (8) %B sinh(x)sinh(y) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 78 +ff:=sinhsinhrule ee +--R +--R 3a x a x 3a x a x +--R - sinh(----) - sinh(---) - cosh(----) - 3cosh(---) +--R 2 2 2 2 +--R (9) ----------------------------------------------------------------- +--R 3a x a x a x a x +--R a sinh(----) + a sinh(---) + 2a cosh(---)cosh(a x) + 2a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 79 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %BC cosh(y + x) + %BC cosh(y - x) +--I (10) %BC cosh(x)cosh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 80 14:575 Schaums and Axiom differ by a constant +gg:=coshcoshrule ff +--R +--R 1 +--R (11) - - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.576~~~~~$\displaystyle @@ -314,7 +1013,7 @@ $$ <<*>>= )clear all ---S 15 of 28 +--S 81 aa:=integrate(1/(cosh(a*x)-1),x) --R --R @@ -323,6 +1022,112 @@ aa:=integrate(1/(cosh(a*x)-1),x) --R a sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 82 +bb:=-1/a*coth((a*x)/2) +--R +--R a x +--R coth(---) +--R 2 +--R (2) - --------- +--R a +--R Type: Expression Integer +--E + +--S 83 +cc:=aa-bb +--R +--R a x a x +--R coth(---)sinh(a x) + (cosh(a x) - 1)coth(---) - 2 +--R 2 2 +--R (3) ------------------------------------------------- +--R a sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 84 +cothrule:=rule(coth(x) == cosh(x)/sinh(x)) +--R +--R cosh(x) +--R (4) coth(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 85 +dd:=cothrule cc +--R +--R a x a x a x a x +--R cosh(---)sinh(a x) - 2sinh(---) + cosh(---)cosh(a x) - cosh(---) +--R 2 2 2 2 +--R (5) ---------------------------------------------------------------- +--R a x a x +--R a sinh(---)sinh(a x) + (a cosh(a x) - a)sinh(---) +--R 2 2 +--R Type: Expression Integer +--E + +--S 86 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %BD sinh(y + x) - %BD sinh(y - x) +--I (6) %BD cosh(y)sinh(x) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 87 +ee:=sinhcoshrule dd +--R +--R 3a x a x a x a x +--R sinh(----) - 3sinh(---) + 2cosh(---)cosh(a x) - 2cosh(---) +--R 2 2 2 2 +--R (7) ---------------------------------------------------------- +--R 3a x a x a x +--R a sinh(----) + 2a sinh(---)sinh(a x) - 3a sinh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 88 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %BE cosh(y + x) - %BE cosh(y - x) +--I (8) %BE sinh(x)sinh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 89 +ff:=sinhsinhrule ee +--R +--R 3a x a x a x a x +--R sinh(----) - 3sinh(---) + 2cosh(---)cosh(a x) - 2cosh(---) +--R 2 2 2 2 +--R (9) ---------------------------------------------------------- +--R 3a x a x 3a x a x +--R a sinh(----) - 3a sinh(---) + a cosh(----) - a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 90 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %BF cosh(y + x) + %BF cosh(y - x) +--I (10) %BF cosh(x)cosh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 91 14:576 Schaums and Axiom differ by a constant +gg:=coshcoshrule ff +--R +--R 1 +--R (11) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.577~~~~~$\displaystyle @@ -333,7 +1138,7 @@ $$ <<*>>= )clear all ---S 16 of 28 +--S 92 aa:=integrate(x/(cosh(a*x)+1),x) --R --R @@ -346,6 +1151,181 @@ aa:=integrate(x/(cosh(a*x)+1),x) --R a sinh(a x) + a cosh(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 93 +bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2)) +--R +--R a x a x +--R - 2log(cosh(---)) + a x tanh(---) +--R 2 2 +--R (2) --------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 94 +cc:=aa-bb +--R +--R (3) +--R (- 2sinh(a x) - 2cosh(a x) - 2)log(sinh(a x) + cosh(a x) + 1) +--R + +--R a x +--R (2sinh(a x) + 2cosh(a x) + 2)log(cosh(---)) +--R 2 +--R + +--R a x +--R (- a x sinh(a x) - a x cosh(a x) - a x)tanh(---) + 2a x sinh(a x) +--R 2 +--R + +--R 2a x cosh(a x) +--R / +--R 2 2 2 +--R a sinh(a x) + a cosh(a x) + a +--R Type: Expression Integer +--E + +--S 95 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (4) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 96 +dd:=tanhrule cc +--R +--R (5) +--R a x a x a x +--R (- 2cosh(---)sinh(a x) - 2cosh(---)cosh(a x) - 2cosh(---)) +--R 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R a x a x a x a x +--R (2cosh(---)sinh(a x) + 2cosh(---)cosh(a x) + 2cosh(---))log(cosh(---)) +--R 2 2 2 2 +--R + +--R a x a x +--R (- a x sinh(---) + 2a x cosh(---))sinh(a x) +--R 2 2 +--R + +--R a x a x +--R (- a x cosh(a x) - a x)sinh(---) + 2a x cosh(---)cosh(a x) +--R 2 2 +--R / +--R 2 a x 2 a x 2 a x +--R a cosh(---)sinh(a x) + a cosh(---)cosh(a x) + a cosh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 97 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %BG cosh(y + x) + %BG cosh(y - x) +--I (6) %BG cosh(x)cosh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 98 +ee:=coshcoshrule dd +--R +--R (7) +--R a x 3a x a x +--R (- 4cosh(---)sinh(a x) - 2cosh(----) - 6cosh(---)) +--R 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R a x 3a x a x a x +--R (4cosh(---)sinh(a x) + 2cosh(----) + 6cosh(---))log(cosh(---)) +--R 2 2 2 2 +--R + +--R a x a x +--R (- 2a x sinh(---) + 4a x cosh(---))sinh(a x) +--R 2 2 +--R + +--R a x 3a x a x +--R (- 2a x cosh(a x) - 2a x)sinh(---) + 2a x cosh(----) + 2a x cosh(---) +--R 2 2 2 +--R / +--R 2 a x 2 3a x 2 a x +--R 2a cosh(---)sinh(a x) + a cosh(----) + 3a cosh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 99 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %BH sinh(y + x) - %BH sinh(y - x) +--I (8) %BH cosh(y)sinh(x) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 100 +ff:=sinhcoshrule ee +--R +--R (9) +--R 3a x a x 3a x a x +--R (- 2sinh(----) - 2sinh(---) - 2cosh(----) - 6cosh(---)) +--R 2 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 3a x a x 3a x a x a x +--R (2sinh(----) + 2sinh(---) + 2cosh(----) + 6cosh(---))log(cosh(---)) +--R 2 2 2 2 2 +--R + +--R 3a x a x a x +--R a x sinh(----) - 2a x sinh(---)sinh(a x) + a x sinh(---) +--R 2 2 2 +--R + +--R 3a x a x +--R 2a x cosh(----) + 2a x cosh(---) +--R 2 2 +--R / +--R 2 3a x 2 a x 2 3a x 2 a x +--R a sinh(----) + a sinh(---) + a cosh(----) + 3a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 101 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %BI cosh(y + x) - %BI cosh(y - x) +--I (10) %BI sinh(x)sinh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 102 +gg:=sinhsinhrule ff +--R +--R a x +--R - 2log(sinh(a x) + cosh(a x) + 1) + 2log(cosh(---)) + a x +--R 2 +--R (11) --------------------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 103 14:577 Schaums and Axiom differ by a constant +complexNormalize gg +--R +--R 2log(2) +--R (12) - ------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.578~~~~~$\displaystyle @@ -356,7 +1336,7 @@ $$ <<*>>= )clear all ---S 17 of 28 +--S 104 aa:=integrate(x/(cosh(a*x)-1),x) --R --R @@ -369,6 +1349,181 @@ aa:=integrate(x/(cosh(a*x)-1),x) --R a sinh(a x) + a cosh(a x) - a --R Type: Union(Expression Integer,...) --E + +--S 105 +bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2)) +--R +--R a x a x +--R 2log(sinh(---)) - a x coth(---) +--R 2 2 +--R (2) ------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 106 +cc:=aa-bb +--R +--R (3) +--R (2sinh(a x) + 2cosh(a x) - 2)log(sinh(a x) + cosh(a x) - 1) +--R + +--R a x +--R (- 2sinh(a x) - 2cosh(a x) + 2)log(sinh(---)) +--R 2 +--R + +--R a x a x +--R (a x coth(---) - 2a x)sinh(a x) + (a x cosh(a x) - a x)coth(---) +--R 2 2 +--R + +--R - 2a x cosh(a x) +--R / +--R 2 2 2 +--R a sinh(a x) + a cosh(a x) - a +--R Type: Expression Integer +--E + +--S 107 +cothrule:=rule(coth(x) == cosh(x)/sinh(x)) +--R +--R cosh(x) +--R (4) coth(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 108 +dd:=cothrule cc +--R +--R (5) +--R a x a x +--R (2sinh(---)sinh(a x) + (2cosh(a x) - 2)sinh(---)) +--R 2 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R a x a x a x +--R (- 2sinh(---)sinh(a x) + (- 2cosh(a x) + 2)sinh(---))log(sinh(---)) +--R 2 2 2 +--R + +--R a x a x a x +--R (- 2a x sinh(---) + a x cosh(---))sinh(a x) - 2a x cosh(a x)sinh(---) +--R 2 2 2 +--R + +--R a x a x +--R a x cosh(---)cosh(a x) - a x cosh(---) +--R 2 2 +--R / +--R 2 a x 2 2 a x +--R a sinh(---)sinh(a x) + (a cosh(a x) - a )sinh(---) +--R 2 2 +--R Type: Expression Integer +--E + +--S 109 +sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %BJ sinh(y + x) - %BJ sinh(y - x) +--I (6) %BJ cosh(y)sinh(x) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 110 +ee:=sinhcoshrule dd +--R +--R (7) +--R 3a x a x a x +--R (2sinh(----) + 4sinh(---)sinh(a x) - 6sinh(---)) +--R 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 3a x a x a x a x +--R (- 2sinh(----) - 4sinh(---)sinh(a x) + 6sinh(---))log(sinh(---)) +--R 2 2 2 2 +--R + +--R 3a x a x a x +--R - a x sinh(----) - 4a x sinh(---)sinh(a x) + 3a x sinh(---) +--R 2 2 2 +--R + +--R a x a x +--R 2a x cosh(---)cosh(a x) - 2a x cosh(---) +--R 2 2 +--R / +--R 2 3a x 2 a x 2 a x +--R a sinh(----) + 2a sinh(---)sinh(a x) - 3a sinh(---) +--R 2 2 2 +--R Type: Expression Integer +--E + +--S 111 +sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %BK cosh(y + x) - %BK cosh(y - x) +--I (8) %BK sinh(x)sinh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 112 +ff:=sinhsinhrule ee +--R +--R (9) +--R 3a x a x 3a x a x +--R (2sinh(----) - 6sinh(---) + 2cosh(----) - 2cosh(---)) +--R 2 2 2 2 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 3a x a x 3a x a x a x +--R (- 2sinh(----) + 6sinh(---) - 2cosh(----) + 2cosh(---))log(sinh(---)) +--R 2 2 2 2 2 +--R + +--R 3a x a x 3a x +--R - a x sinh(----) + 3a x sinh(---) - 2a x cosh(----) +--R 2 2 2 +--R + +--R a x +--R 2a x cosh(---)cosh(a x) +--R 2 +--R / +--R 2 3a x 2 a x 2 3a x 2 a x +--R a sinh(----) - 3a sinh(---) + a cosh(----) - a cosh(---) +--R 2 2 2 2 +--R Type: Expression Integer +--E + +--S 113 +coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %BL cosh(y + x) + %BL cosh(y - x) +--I (10) %BL cosh(x)cosh(y) == --------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 114 +gg:=coshcoshrule ff +--R +--R a x +--R 2log(sinh(a x) + cosh(a x) - 1) - 2log(sinh(---)) - a x +--R 2 +--R (11) ------------------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 115 14:578 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 2log(2) +--R (12) ------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.579~~~~~$\displaystyle @@ -379,7 +1534,7 @@ $$ <<*>>= )clear all ---S 18 of 28 +--S 116 aa:=integrate(1/(cosh(a*x)+1)^2,x) --R --R @@ -396,6 +1551,59 @@ aa:=integrate(1/(cosh(a*x)+1)^2,x) --R 9a cosh(a x) + 9a cosh(a x) + 3a --R Type: Union(Expression Integer,...) --E + +--S 117 +bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3 +--R +--R a x 3 a x +--R - tanh(---) + 3tanh(---) +--R 2 2 +--R (2) ------------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 118 14:579 Axiom cannot compute this integral +cc:=aa-bb +--R +--R (3) +--R 3 2 +--R sinh(a x) + (3cosh(a x) + 3)sinh(a x) +--R + +--R 2 3 2 +--R (3cosh(a x) + 6cosh(a x) + 3)sinh(a x) + cosh(a x) + 3cosh(a x) +--R + +--R 3cosh(a x) + 1 +--R * +--R a x 3 +--R tanh(---) +--R 2 +--R + +--R 3 2 +--R - 3sinh(a x) + (- 9cosh(a x) - 9)sinh(a x) +--R + +--R 2 3 +--R (- 9cosh(a x) - 18cosh(a x) - 9)sinh(a x) - 3cosh(a x) +--R + +--R 2 +--R - 9cosh(a x) - 9cosh(a x) - 3 +--R * +--R a x +--R tanh(---) +--R 2 +--R + +--R - 12sinh(a x) - 12cosh(a x) - 4 +--R / +--R 3 2 +--R 6a sinh(a x) + (18a cosh(a x) + 18a)sinh(a x) +--R + +--R 2 3 +--R (18a cosh(a x) + 36a cosh(a x) + 18a)sinh(a x) + 6a cosh(a x) +--R + +--R 2 +--R 18a cosh(a x) + 18a cosh(a x) + 6a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.580~~~~~$\displaystyle @@ -406,7 +1614,7 @@ $$ <<*>>= )clear all ---S 19 of 28 +--S 119 aa:=integrate(1/(cosh(a*x)-1)^2,x) --R --R @@ -423,6 +1631,58 @@ aa:=integrate(1/(cosh(a*x)-1)^2,x) --R - 9a cosh(a x) + 9a cosh(a x) - 3a --R Type: Union(Expression Integer,...) --E + +--S 120 +bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3 +--R +--R a x 3 a x +--R - coth(---) + 3coth(---) +--R 2 2 +--R (2) ------------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 121 14:580 Axiom cannot compute this integral +cc:=aa-bb +--R +--R (3) +--R a x 3 a x 3 +--R (coth(---) - 3coth(---))sinh(a x) +--R 2 2 +--R + +--R a x 3 a x 2 +--R ((3cosh(a x) - 3)coth(---) + (- 9cosh(a x) + 9)coth(---))sinh(a x) +--R 2 2 +--R + +--R 2 a x 3 +--R (3cosh(a x) - 6cosh(a x) + 3)coth(---) +--R 2 +--R + +--R 2 a x +--R (- 9cosh(a x) + 18cosh(a x) - 9)coth(---) - 12 +--R 2 +--R * +--R sinh(a x) +--R + +--R 3 2 a x 3 +--R (cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1)coth(---) +--R 2 +--R + +--R 3 2 a x +--R (- 3cosh(a x) + 9cosh(a x) - 9cosh(a x) + 3)coth(---) - 12cosh(a x) + 4 +--R 2 +--R / +--R 3 2 +--R 6a sinh(a x) + (18a cosh(a x) - 18a)sinh(a x) +--R + +--R 2 3 +--R (18a cosh(a x) - 36a cosh(a x) + 18a)sinh(a x) + 6a cosh(a x) +--R + +--R 2 +--R - 18a cosh(a x) + 18a cosh(a x) - 6a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.581~~~~~$\displaystyle @@ -442,7 +1702,7 @@ $$ <<*>>= )clear all ---S 20 of 28 +--S 122 aa:=integrate(1/(p+q*cosh(a*x)),x) --R --R @@ -483,6 +1743,159 @@ aa:=integrate(1/(p+q*cosh(a*x)),x) --R a\|q - p --R Type: Union(List Expression Integer,...) --E + +--S 123 +bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2)) +--R +--R a x +--R q %e + p +--R 2atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R (2) ------------------ +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 124 +bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2-q^2))) +--R +--R +---------+ +--R | 2 2 a x +--R - \|- q + p + q %e + p +--R log(----------------------------) +--R +---------+ +--R | 2 2 a x +--R \|- q + p + q %e + p +--R (3) --------------------------------- +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 125 +cc1:=aa.1-bb1 +--R +--R (4) +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ a x +--R | 2 2 q %e + p +--R - 2\|- q + p atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 126 +cc2:=aa.2-bb1 +--R +--R +-------+ +--R | 2 2 a x +--R (q sinh(a x) + q cosh(a x) + p)\|q - p q %e + p +--R 2atan(-----------------------------------------) - 2atan(-----------) +--R 2 2 +-------+ +--R q - p | 2 2 +--R \|q - p +--R (5) --------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 127 +cc3:=aa.1-bb2 +--R +--R (6) +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 a x +--R - \|- q + p + q %e + p +--R - log(----------------------------) +--R +---------+ +--R | 2 2 a x +--R \|- q + p + q %e + p +--R / +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 128 14:581 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (7) +--R +---------+ +--R +-------+ | 2 2 a x +--R | 2 2 - \|- q + p + q %e + p +--R - \|q - p log(----------------------------) +--R +---------+ +--R | 2 2 a x +--R \|- q + p + q %e + p +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2\|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.582~~~~~$\displaystyle @@ -494,7 +1907,7 @@ $$ <<*>>= )clear all ---S 21 of 28 +--S 129 aa:=integrate(1/(p+q*cosh(a*x))^2,x) --R --R @@ -575,6 +1988,408 @@ aa:=integrate(1/(p+q*cosh(a*x))^2,x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 130 +t1:=integrate(1/(p+q*cosh(a*x)),x) +--R +--R (2) +--R [ +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R / +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R , +--R +-------+ +--R | 2 2 +--R (q sinh(a x) + q cosh(a x) + p)\|q - p +--R 2atan(-----------------------------------------) +--R 2 2 +--R q - p +--R ------------------------------------------------] +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Union(List Expression Integer,...) +--E + +--S 131 +bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1 +--R +--R (3) +--R 2 +--R (- p q cosh(a x) - p ) +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R q sinh(a x)\|- q + p +--R / +--R +---------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cosh(a x) + a p q - a p )\|- q + p +--R Type: Expression Integer +--E + +--S 132 +bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2 +--R +--R (4) +--R +-------+ +--R | 2 2 +--R 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R (- 2p q cosh(a x) - 2p )atan(-----------------------------------------) +--R 2 2 +--R q - p +--R + +--R +-------+ +--R | 2 2 +--R q sinh(a x)\|q - p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cosh(a x) + a p q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 133 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 2 2 +--R (p q cosh(a x) + p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R 2 2 2 +--R (p q cosh(a x) + p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R 2 3 2 2 +--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) +--R + +--R 2 2 2 +--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R / +--R 4 2 2 3 3 2 +--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x) +--R + +--R 4 2 2 2 3 3 2 2 +--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p q +--R + +--R 4 +--R - 2a p +--R * +--R sinh(a x) +--R + +--R 4 2 2 3 3 3 2 +--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x) +--R + +--R 4 2 2 4 3 3 +--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R Type: Expression Integer +--E + +--S 134 +cc2:=aa.2-bb1 +--R +--R (6) +--R 2 2 2 +--R (p q cosh(a x) + p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R 2 2 2 +--R (- 2p q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (- 4p q cosh(a x) - 8p q cosh(a x) - 4p )sinh(a x) - 2p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R - 6p q cosh(a x) + (- 2p q - 4p )cosh(a x) - 2p q +--R * +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R \|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R + +--R 2 3 2 2 +--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) +--R + +--R 2 2 2 +--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q +--R * +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R \|- q + p \|q - p +--R / +--R 4 2 2 3 3 2 +--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x) +--R + +--R 4 2 2 2 3 3 2 2 +--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p q +--R + +--R 4 +--R - 2a p +--R * +--R sinh(a x) +--R + +--R 4 2 2 3 3 3 2 +--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x) +--R + +--R 4 2 2 4 3 3 +--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q +--R * +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R \|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 135 +cc3:=aa.1-bb2 +--R +--R (7) +--R 2 2 2 +--R (p q cosh(a x) + p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R 2 2 2 +--R (2p q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 3 2 3 +--R (4p q cosh(a x) + 8p q cosh(a x) + 4p )sinh(a x) + 2p q cosh(a x) +--R + +--R 2 2 2 3 2 +--R 6p q cosh(a x) + (2p q + 4p )cosh(a x) + 2p q +--R * +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p +--R \|- q + p atan(-----------------------------------------) +--R 2 2 +--R q - p +--R + +--R 2 3 2 2 +--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) +--R + +--R 2 2 2 +--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q +--R * +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R \|- q + p \|q - p +--R / +--R 4 2 2 3 3 2 +--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x) +--R + +--R 4 2 2 2 3 3 2 2 +--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p q +--R + +--R 4 +--R - 2a p +--R * +--R sinh(a x) +--R + +--R 4 2 2 3 3 3 2 +--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x) +--R + +--R 4 2 2 4 3 3 +--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q +--R * +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R \|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 136 14:582 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (8) +--R 2 3 2 2 +--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) - 2p q cosh(a x) +--R + +--R 2 2 +--R (- 2q - 2p )cosh(a x) - 2p q +--R / +--R 4 2 2 3 3 2 +--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x) +--R + +--R 4 2 2 2 3 3 2 2 +--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p q +--R + +--R 4 +--R - 2a p +--R * +--R sinh(a x) +--R + +--R 4 2 2 3 3 3 2 +--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x) +--R + +--R 4 2 2 4 3 3 +--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q +--R Type: Expression Integer +--E @ \section{\cite{1}:14.583~~~~~$\displaystyle @@ -594,7 +2409,7 @@ $$ <<*>>= )clear all ---S 22 of 28 +--S 137 aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x) --R --R @@ -661,6 +2476,219 @@ aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 138 +bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p^2-q^2))) +--R +--R +---------+ +--R | 2 2 +--R - \|- q + p - p tanh(a x) +--R log(----------------------------) +--R +---------+ +--R | 2 2 +--R \|- q + p - p tanh(a x) +--R (2) --------------------------------- +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R Type: Expression Integer +--E + +--S 139 +bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2)) +--R +--R p tanh(a x) +--R atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R (3) - ----------------- +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + +--S 140 +cc1:=aa.1-bb1 +--R +--R (4) +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q - 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q - 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q - 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 4 3 2 +--R (- 4p q + 4p q )sinh(a x) + (- 8p q + 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q + 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q - 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q - 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q - 4p )cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R - \|- q + p - p tanh(a x) +--R - log(----------------------------) +--R +---------+ +--R | 2 2 +--R \|- q + p - p tanh(a x) +--R / +--R +---------+ +--R | 2 2 +--R 2a p\|- q + p +--R Type: Expression Integer +--E + +--S 141 +cc2:=aa.2-bb1 +--R +--R (5) +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 - \|- q + p - p tanh(a x) +--R - \|q - p log(----------------------------) +--R +---------+ +--R | 2 2 +--R \|- q + p - p tanh(a x) +--R + +--R - +--R +---------+ +--R | 2 2 +--R 2\|- q + p +--R * +--R atan +--R 2 2 2 2 2 2 +--R q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) + q +--R + +--R 2 +--R - 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 142 +cc3:=aa.1-bb2 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R \|q - p +--R * +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q - 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q - 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q - 4p q )cosh(a x) + q - 8p q + 8p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 4 3 2 2 +--R (- 4p q + 4p q )sinh(a x) +--R + +--R 4 3 2 +--R (- 8p q + 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q + 4p q )cosh(a x) - 4p q + 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q - 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q - 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q - 4p )cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 p tanh(a x) +--R 2\|- q + p atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 143 14:583 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (7) +--R - +--R atan +--R 2 2 2 2 2 2 +--R q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) + q +--R + +--R 2 +--R - 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R 2 3 +--R 2p q - 2p +--R + +--R p tanh(a x) +--R atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q - p +--R / +--R +-------+ +--R | 2 2 +--R a p\|q - p +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.584~~~~~$\displaystyle @@ -680,7 +2708,7 @@ $$ <<*>>= )clear all ---S 23 of 28 +--S 144 aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x) --R --R @@ -725,6 +2753,142 @@ aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x) --R 2a p\|q + p --R Type: Union(Expression Integer,...) --E + +--S 145 +bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p^2+q^2))) +--R +--R +-------+ +--R | 2 2 +--R - \|q + p - p tanh(a x) +--R log(--------------------------) +--R +-------+ +--R | 2 2 +--R \|q + p - p tanh(a x) +--R (2) ------------------------------- +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Expression Integer +--E + +--S 146 +bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2)) +--R +--R p tanh(a x) +--R atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q + p +--R (3) ----------------- +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Expression Integer +--E + +--S 147 +cc1:=aa-bb1 +--R +--R (4) +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q + 4p q )cosh(a x) + q + 8p q + 8p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 4 3 2 2 4 3 2 +--R (- 4p q - 4p q )sinh(a x) + (- 8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q - 4p q )cosh(a x) - 4p q - 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q + 4p )cosh(a x) + q +--R + +--R +-------+ +--R | 2 2 +--R - \|q + p - p tanh(a x) +--R - log(--------------------------) +--R +-------+ +--R | 2 2 +--R \|q + p - p tanh(a x) +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Expression Integer +--E + +--S 148 14:584 Axiom cannot compute this integral +cc2:=aa-bb2 +--R +--R (5) +--R log +--R 4 4 4 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 4 2 4 2 2 2 +--R (6q cosh(a x) + 2q + 4p q )sinh(a x) +--R + +--R 4 3 4 2 2 +--R (4q cosh(a x) + (4q + 8p q )cosh(a x))sinh(a x) +--R + +--R 4 4 4 2 2 2 4 2 2 4 +--R q cosh(a x) + (2q + 4p q )cosh(a x) + q + 8p q + 8p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 4 3 2 2 4 3 2 +--R (- 4p q - 4p q )sinh(a x) + (- 8p q - 8p q )cosh(a x)sinh(a x) +--R + +--R 4 3 2 2 4 3 2 5 +--R (- 4p q - 4p q )cosh(a x) - 4p q - 12p q - 8p +--R / +--R 2 4 2 3 +--R q sinh(a x) + 4q cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (6q cosh(a x) + 2q + 4p )sinh(a x) +--R + +--R 2 3 2 2 2 4 +--R (4q cosh(a x) + (4q + 8p )cosh(a x))sinh(a x) + q cosh(a x) +--R + +--R 2 2 2 2 +--R (2q + 4p )cosh(a x) + q +--R + +--R p tanh(a x) +--R - 2atan(-----------) +--R +-------+ +--R | 2 2 +--R \|q + p +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q + p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.585~~~~~$\displaystyle @@ -735,7 +2899,7 @@ $$ <<*>>= )clear all ---S 24 of 28 +--S 149 14:585 Axiom cannot compute this integral aa:=integrate(x^m*cosh(a*x),x) --R --R @@ -755,7 +2919,7 @@ $$ <<*>>= )clear all ---S 25 of 28 +--S 150 14:586 Axiom cannot compute this integral aa:=integrate(cosh(a*x)^n,x) --R --R @@ -776,7 +2940,7 @@ $$ <<*>>= )clear all ---S 26 of 28 +--S 151 14:587 Axiom cannot compute this integral aa:=integrate(cosh(a*x)/x^n,x) --R --R @@ -798,7 +2962,7 @@ $$ <<*>>= )clear all ---S 27 of 28 +--S 152 14:588 Axiom cannot compute this integral aa:=integrate(1/cosh(a*x)^n,x) --R --R @@ -821,7 +2985,7 @@ $$ <<*>>= )clear all ---S 28 of 28 +--S 153 14:589 Axiom cannot compute this integral aa:=integrate(1/cosh(a*x)^n,x) --R --R