\(\int \frac {x \cosh (c+d x)}{(a+b x^3)^3} \, dx\) [111]

Optimal result

Integrand size = 17, antiderivative size = 1147 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Output:

-1/18*cosh(d*x+c)/a/b^2/x^4+2/9*cosh(d*x+c)/a^2/b/x-1/6*cosh(d*x+c)/b/x/(b 
*x^3+a)^2+1/18*cosh(d*x+c)/b^2/x^4/(b*x^3+a)-2/27*(-1)^(2/3)*cosh(c+(-1)^( 
1/3)*a^(1/3)*d/b^(1/3))*Chi((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)/a^(7/3)/b^(2 
/3)+1/54*(-1)^(1/3)*d^2*cosh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))*Chi((-1)^(1/3 
)*a^(1/3)*d/b^(1/3)-d*x)/a^(5/3)/b^(4/3)+2/27*(-1)^(1/3)*cosh(c-(-1)^(2/3) 
*a^(1/3)*d/b^(1/3))*Chi(-(-1)^(2/3)*a^(1/3)*d/b^(1/3)-d*x)/a^(7/3)/b^(2/3) 
-1/54*(-1)^(2/3)*d^2*cosh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Chi(-(-1)^(2/3)* 
a^(1/3)*d/b^(1/3)-d*x)/a^(5/3)/b^(4/3)-2/27*cosh(c-a^(1/3)*d/b^(1/3))*Chi( 
a^(1/3)*d/b^(1/3)+d*x)/a^(7/3)/b^(2/3)-1/54*d^2*cosh(c-a^(1/3)*d/b^(1/3))* 
Chi(a^(1/3)*d/b^(1/3)+d*x)/a^(5/3)/b^(4/3)-2/27*d*Chi(a^(1/3)*d/b^(1/3)+d* 
x)*sinh(c-a^(1/3)*d/b^(1/3))/a^2/b-2/27*d*Chi((-1)^(1/3)*a^(1/3)*d/b^(1/3) 
-d*x)*sinh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^2/b-2/27*d*Chi(-(-1)^(2/3)*a^ 
(1/3)*d/b^(1/3)-d*x)*sinh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^2/b+1/18*d*sin 
h(d*x+c)/a/b^2/x^3-1/18*d*sinh(d*x+c)/b^2/x^3/(b*x^3+a)-2/27*d*cosh(c+(-1) 
^(1/3)*a^(1/3)*d/b^(1/3))*Shi(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/a^2/b-2/2 
7*(-1)^(2/3)*sinh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))*Shi(-(-1)^(1/3)*a^(1/3)* 
d/b^(1/3)+d*x)/a^(7/3)/b^(2/3)+1/54*(-1)^(1/3)*d^2*sinh(c+(-1)^(1/3)*a^(1/ 
3)*d/b^(1/3))*Shi(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(5/3)/b^(4/3)-2/27* 
d*cosh(c-a^(1/3)*d/b^(1/3))*Shi(a^(1/3)*d/b^(1/3)+d*x)/a^2/b-2/27*sinh(c-a 
^(1/3)*d/b^(1/3))*Shi(a^(1/3)*d/b^(1/3)+d*x)/a^(7/3)/b^(2/3)-1/54*d^2*s...
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 0.31 (sec) , antiderivative size = 669, normalized size of antiderivative = 0.58 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(x*Cosh[c + d*x])/(a + b*x^3)^3,x]
 

Output:

(RootSum[a + b*#1^3 & , (-(a*d^2*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]) 
+ a*d^2*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] + a*d^2*Cosh[c + d*#1]*Sin 
hIntegral[d*(x - #1)] - a*d^2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + 4* 
b*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]*#1 - 4*b*CoshIntegral[d*(x - #1) 
]*Sinh[c + d*#1]*#1 - 4*b*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 + 4*b 
*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 + 4*b*d*Cosh[c + d*#1]*CoshInt 
egral[d*(x - #1)]*#1^2 - 4*b*d*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1^ 
2 - 4*b*d*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1^2 + 4*b*d*Sinh[c + d* 
#1]*SinhIntegral[d*(x - #1)]*#1^2)/#1^2 & ] - RootSum[a + b*#1^3 & , (a*d^ 
2*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] + a*d^2*CoshIntegral[d*(x - #1)] 
*Sinh[c + d*#1] + a*d^2*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] + a*d^2*Si 
nh[c + d*#1]*SinhIntegral[d*(x - #1)] - 4*b*Cosh[c + d*#1]*CoshIntegral[d* 
(x - #1)]*#1 - 4*b*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1 - 4*b*Cosh[c 
 + d*#1]*SinhIntegral[d*(x - #1)]*#1 - 4*b*Sinh[c + d*#1]*SinhIntegral[d*( 
x - #1)]*#1 + 4*b*d*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]*#1^2 + 4*b*d*C 
oshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1^2 + 4*b*d*Cosh[c + d*#1]*SinhInt 
egral[d*(x - #1)]*#1^2 + 4*b*d*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1^ 
2)/#1^2 & ] + (6*b*Cosh[d*x]*(b*x^2*(7*a + 4*b*x^3)*Cosh[c] + a*d*(a + b*x 
^3)*Sinh[c]))/(a + b*x^3)^2 + (6*b*(a*d*(a + b*x^3)*Cosh[c] + b*x^2*(7*a + 
 4*b*x^3)*Sinh[c])*Sinh[d*x])/(a + b*x^3)^2)/(108*a^2*b^2)
 

Rubi [A] (verified)

Time = 4.97 (sec) , antiderivative size = 1813, normalized size of antiderivative = 1.58, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5814, 5813, 5814, 5815, 2009, 5816, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx\)

\(\Big \downarrow \) 5814

\(\displaystyle \frac {d \int \frac {\sinh (c+d x)}{x \left (b x^3+a\right )^2}dx}{6 b}-\frac {\int \frac {\cosh (c+d x)}{x^2 \left (b x^3+a\right )^2}dx}{6 b}-\frac {\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 5813

\(\displaystyle \frac {d \left (\frac {d \int \frac {\cosh (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\int \frac {\sinh (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}-\frac {\sinh (c+d x)}{3 b x^3 \left (a+b x^3\right )}\right )}{6 b}-\frac {\int \frac {\cosh (c+d x)}{x^2 \left (b x^3+a\right )^2}dx}{6 b}-\frac {\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 5814

\(\displaystyle \frac {d \left (\frac {d \int \frac {\cosh (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\int \frac {\sinh (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}-\frac {\sinh (c+d x)}{3 b x^3 \left (a+b x^3\right )}\right )}{6 b}-\frac {-\frac {4 \int \frac {\cosh (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}+\frac {d \int \frac {\sinh (c+d x)}{x^4 \left (b x^3+a\right )}dx}{3 b}-\frac {\cosh (c+d x)}{3 b x^4 \left (a+b x^3\right )}}{6 b}-\frac {\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 5815

\(\displaystyle \frac {d \left (-\frac {\int \left (\frac {b^2 \sinh (c+d x) x^2}{a^2 \left (b x^3+a\right )}-\frac {b \sinh (c+d x)}{a^2 x}+\frac {\sinh (c+d x)}{a x^4}\right )dx}{b}+\frac {d \int \frac {\cosh (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\sinh (c+d x)}{3 b x^3 \left (a+b x^3\right )}\right )}{6 b}-\frac {\frac {d \int \left (\frac {b^2 \sinh (c+d x) x^2}{a^2 \left (b x^3+a\right )}-\frac {b \sinh (c+d x)}{a^2 x}+\frac {\sinh (c+d x)}{a x^4}\right )dx}{3 b}-\frac {4 \int \frac {\cosh (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}-\frac {\cosh (c+d x)}{3 b x^4 \left (a+b x^3\right )}}{6 b}-\frac {\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\cosh (c+d x)}{6 b x \left (b x^3+a\right )^2}-\frac {-\frac {\cosh (c+d x)}{3 b x^4 \left (b x^3+a\right )}+\frac {d \left (\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}\right )}{3 b}-\frac {4 \int \frac {\cosh (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}}{6 b}+\frac {d \left (-\frac {\sinh (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}}{b}+\frac {d \int \frac {\cosh (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}\right )}{6 b}\)

\(\Big \downarrow \) 5816

\(\displaystyle -\frac {\cosh (c+d x)}{6 b x \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\sinh (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}}{b}+\frac {d \int \left (\frac {\cosh (c+d x)}{a x^3}-\frac {b \cosh (c+d x)}{a \left (b x^3+a\right )}\right )dx}{3 b}\right )}{6 b}-\frac {-\frac {\cosh (c+d x)}{3 b x^4 \left (b x^3+a\right )}+\frac {d \left (\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}\right )}{3 b}-\frac {4 \int \left (\frac {x \cosh (c+d x) b^2}{a^2 \left (b x^3+a\right )}-\frac {\cosh (c+d x) b}{a^2 x^2}+\frac {\cosh (c+d x)}{a x^5}\right )dx}{3 b}}{6 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\cosh (c+d x)}{6 b x \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\sinh (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}}{b}+\frac {d \left (\frac {\cosh (c) \text {Chi}(d x) d^2}{2 a}+\frac {\sinh (c) \text {Shi}(d x) d^2}{2 a}-\frac {\sinh (c+d x) d}{2 a x}-\frac {\cosh (c+d x)}{2 a x^2}+\frac {\sqrt [3]{-1} b^{2/3} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {b^{2/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {\sqrt [3]{-1} b^{2/3} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}\right )}{3 b}\right )}{6 b}-\frac {-\frac {\cosh (c+d x)}{3 b x^4 \left (b x^3+a\right )}+\frac {d \left (\frac {\cosh (c) \text {Chi}(d x) d^3}{6 a}+\frac {\sinh (c) \text {Shi}(d x) d^3}{6 a}-\frac {\sinh (c+d x) d^2}{6 a x}-\frac {\cosh (c+d x) d}{6 a x^2}-\frac {b \text {Chi}(d x) \sinh (c)}{a^2}+\frac {b \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sinh (c+d x)}{3 a x^3}-\frac {b \cosh (c) \text {Shi}(d x)}{a^2}-\frac {b \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}\right )}{3 b}-\frac {4 \left (\frac {\cosh (c) \text {Chi}(d x) d^4}{24 a}+\frac {\sinh (c) \text {Shi}(d x) d^4}{24 a}-\frac {\sinh (c+d x) d^3}{24 a x}-\frac {\cosh (c+d x) d^2}{24 a x^2}-\frac {b \text {Chi}(d x) \sinh (c) d}{a^2}-\frac {\sinh (c+d x) d}{12 a x^3}-\frac {b \cosh (c) \text {Shi}(d x) d}{a^2}+\frac {b \cosh (c+d x)}{a^2 x}-\frac {\cosh (c+d x)}{4 a x^4}-\frac {(-1)^{2/3} b^{4/3} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}-\frac {b^{4/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}+\frac {(-1)^{2/3} b^{4/3} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{7/3}}-\frac {b^{4/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}\right )}{3 b}}{6 b}\)

Input:

Int[(x*Cosh[c + d*x])/(a + b*x^3)^3,x]
 

Output:

-1/6*Cosh[c + d*x]/(b*x*(a + b*x^3)^2) + (d*(-1/3*Sinh[c + d*x]/(b*x^3*(a 
+ b*x^3)) - (-1/6*(d*Cosh[c + d*x])/(a*x^2) + (d^3*Cosh[c]*CoshIntegral[d* 
x])/(6*a) - (b*CoshIntegral[d*x]*Sinh[c])/a^2 + (b*CoshIntegral[(a^(1/3)*d 
)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(3*a^2) + (b*CoshIntegral[ 
((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1 
/3)])/(3*a^2) + (b*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*S 
inh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^2) - Sinh[c + d*x]/(3*a*x^3) 
 - (d^2*Sinh[c + d*x])/(6*a*x) - (b*Cosh[c]*SinhIntegral[d*x])/a^2 + (d^3* 
Sinh[c]*SinhIntegral[d*x])/(6*a) - (b*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1 
/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^2) + (b*Cosh 
[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) 
 + (b*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^ 
(1/3)*d)/b^(1/3) + d*x])/(3*a^2))/b + (d*(-1/2*Cosh[c + d*x]/(a*x^2) + (d^ 
2*Cosh[c]*CoshIntegral[d*x])/(2*a) + ((-1)^(1/3)*b^(2/3)*Cosh[c + ((-1)^(1 
/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x] 
)/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3 
)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a^(5/3)) - (b 
^(2/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d* 
x])/(3*a^(5/3)) - (d*Sinh[c + d*x])/(2*a*x) + (d^2*Sinh[c]*SinhIntegral[d* 
x])/(2*a) - ((-1)^(1/3)*b^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5813

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Sy 
mbol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Sinh[c + d*x]/(b*n*(p + 1) 
)), x] + (-Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*(a + b*x^n)^(p + 
1)*Sinh[c + d*x], x], x] - Simp[d/(b*n*(p + 1))   Int[x^(m - n + 1)*(a + b* 
x^n)^(p + 1)*Cosh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1 
] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
 

rule 5814

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sy 
mbol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Cosh[c + d*x]/(b*n*(p + 1) 
)), x] + (-Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*(a + b*x^n)^(p + 
1)*Cosh[c + d*x], x], x] - Simp[d/(b*n*(p + 1))   Int[x^(m - n + 1)*(a + b* 
x^n)^(p + 1)*Sinh[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1 
] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
 

rule 5815

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Sy 
mbol] :> Int[ExpandIntegrand[Sinh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Fr 
eeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 
2] || EqQ[p, -1])
 

rule 5816

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sy 
mbol] :> Int[ExpandIntegrand[Cosh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Fr 
eeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 
2] || EqQ[p, -1])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.38 (sec) , antiderivative size = 1416, normalized size of antiderivative = 1.23

method	result	size
risch	\(\text {Expression too large to display}\)	\(1416\)

Input:

int(x*cosh(d*x+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/108*(-sum((_R1^2-2*_R1*c+c^2-6*_R1+6*c+10)/(_R1^2-2*_R1*c+c^2)*exp(_R1) 
*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*b^ 
3*c*d*x^6-sum((_R1^2-2*_R1*c+c^2+6*_R1-6*c+10)/(_R1^2-2*_R1*c+c^2)*exp(-_R 
1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*b 
^3*c*d*x^6+sum((_R2^2*b*c-2*_R2*b*c^2-a*d^3+b*c^3-4*_R2^2*b+2*_R2*b*c+2*b* 
c^2+4*_R2*b+6*b*c)/(_R2^2-2*_R2*c+c^2)*exp(_R2)*Ei(1,-d*x+_R2-c),_R2=RootO 
f(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*b^2*d*x^6+sum((_R2^2*b*c-2*_R 
2*b*c^2-a*d^3+b*c^3+4*_R2^2*b-2*_R2*b*c-2*b*c^2+4*_R2*b+6*b*c)/(_R2^2-2*_R 
2*c+c^2)*exp(-_R2)*Ei(1,d*x-_R2+c),_R2=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2 
+a*d^3-b*c^3))*b^2*d*x^6-2*sum((_R1^2-2*_R1*c+c^2-6*_R1+6*c+10)/(_R1^2-2*_ 
R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^ 
2+a*d^3-b*c^3))*a*b^2*c*d*x^3-12*exp(-d*x-c)*b^3*x^5-2*sum((_R1^2-2*_R1*c+ 
c^2+6*_R1-6*c+10)/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf 
(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*a*b^2*c*d*x^3-12*exp(d*x+c)*b^ 
3*x^5+3*a*b^2*d*x^3*exp(-d*x-c)-3*a*b^2*d*x^3*exp(d*x+c)+2*sum((_R2^2*b*c- 
2*_R2*b*c^2-a*d^3+b*c^3-4*_R2^2*b+2*_R2*b*c+2*b*c^2+4*_R2*b+6*b*c)/(_R2^2- 
2*_R2*c+c^2)*exp(_R2)*Ei(1,-d*x+_R2-c),_R2=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b 
*c^2+a*d^3-b*c^3))*a*b*d*x^3+2*sum((_R2^2*b*c-2*_R2*b*c^2-a*d^3+b*c^3+4*_R 
2^2*b-2*_R2*b*c-2*b*c^2+4*_R2*b+6*b*c)/(_R2^2-2*_R2*c+c^2)*exp(-_R2)*Ei(1, 
d*x-_R2+c),_R2=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*a*b*d*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4691 vs. \(2 (843) = 1686\).

Time = 0.19 (sec) , antiderivative size = 4691, normalized size of antiderivative = 4.09 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x*cosh(d*x+c)/(b*x**3+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \] Input:

integrate(x*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="maxima")
 

Output:

1/2*(x*e^(d*x + 2*c) - x*e^(-d*x))/(b^3*d*x^9*e^c + 3*a*b^2*d*x^6*e^c + 3* 
a^2*b*d*x^3*e^c + a^3*d*e^c) + 1/2*integrate((8*b*x^3*e^c - a*e^c)*e^(d*x) 
/(b^4*d*x^12 + 4*a*b^3*d*x^9 + 6*a^2*b^2*d*x^6 + 4*a^3*b*d*x^3 + a^4*d), x 
) - 1/2*integrate((8*b*x^3 - a)*e^(-d*x)/(b^4*d*x^12*e^c + 4*a*b^3*d*x^9*e 
^c + 6*a^2*b^2*d*x^6*e^c + 4*a^3*b*d*x^3*e^c + a^4*d*e^c), x)
 

Giac [F]

\[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \] Input:

integrate(x*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="giac")
 

Output:

integrate(x*cosh(d*x + c)/(b*x^3 + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \] Input:

int((x*cosh(c + d*x))/(a + b*x^3)^3,x)
 

Output:

int((x*cosh(c + d*x))/(a + b*x^3)^3, x)
 

Reduce [F]

\[ \int \frac {x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\frac {e^{2 c} \left (\int \frac {e^{d x} x}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right )+\int \frac {x}{e^{d x} a^{3}+3 e^{d x} a^{2} b \,x^{3}+3 e^{d x} a \,b^{2} x^{6}+e^{d x} b^{3} x^{9}}d x}{2 e^{c}} \] Input:

int(x*cosh(d*x+c)/(b*x^3+a)^3,x)
 

Output:

(e**(2*c)*int((e**(d*x)*x)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x* 
*9),x) + int(x/(e**(d*x)*a**3 + 3*e**(d*x)*a**2*b*x**3 + 3*e**(d*x)*a*b**2 
*x**6 + e**(d*x)*b**3*x**9),x))/(2*e**c)