\(\int \frac {x^4 \cosh (c+d x)}{(a+b x^3)^3} \, dx\) [108]

Optimal result

Integrand size = 19, antiderivative size = 1105 \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.37 (sec) , antiderivative size = 675, normalized size of antiderivative = 0.61 \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(x^4*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]*SinhIn 
tegral[d*(x - #1)] + a*d^2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + 2*b*C 
osh[c + d*#1]*CoshIntegral[d*(x - #1)]*#1 - 2*b*CoshIntegral[d*(x - #1)]*S 
inh[c + d*#1]*#1 - 2*b*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 + 2*b*Si 
nh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 + 2*b*d*Cosh[c + d*#1]*CoshIntegr 
al[d*(x - #1)]*#1^2 - 2*b*d*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1^2 - 
 2*b*d*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1^2 + 2*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)] - 2*b*Cosh[c + d*#1]*CoshIntegral[d* 
(x - #1)]*#1 - 2*b*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1 - 2*b*Cosh[c 
 + d*#1]*SinhIntegral[d*(x - #1)]*#1 - 2*b*Sinh[c + d*#1]*SinhIntegral[d*( 
x - #1)]*#1 + 2*b*d*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]*#1^2 + 2*b*d*C 
oshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1^2 + 2*b*d*Cosh[c + d*#1]*SinhInt 
egral[d*(x - #1)]*#1^2 + 2*b*d*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1^ 
2)/#1^2 & ] + (6*b*Cosh[d*x]*(b*x^2*(-a + 2*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*(-a 
+ 2*b*x^3)*Sinh[c])*Sinh[d*x])/(a + b*x^3)^2)/(108*a*b^3)
 

Rubi [A] (verified)

Time = 3.63 (sec) , antiderivative size = 1142, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {5814, 5811, 5804, 2009, 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.

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

\(\Big \downarrow \) 5814

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

\(\Big \downarrow \) 5811

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

\(\Big \downarrow \) 5804

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5814

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

\(\Big \downarrow \) 5815

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^4*Cosh[c + d*x])/(a + b*x^3)^3,x]
 

Output:

-1/6*(x^2*Cosh[c + d*x])/(b*(a + b*x^3)^2) + (d*(-1/3*Sinh[c + d*x]/(b*(a 
+ b*x^3)) + (d*(-1/3*((-1)^(1/3)*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])/(a^(2/3)*b^(1/3)) + (( 
-1)^(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^(2/3)*b^(1/3)) + (Cosh[c - (a^(1/3)*d 
)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3)) + 
((-1)^(1/3)*Sinh[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^(2/3)*b^(1/3)) + (Sinh[c - (a^(1/3)*d) 
/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3)) + ( 
(-1)^(2/3)*Sinh[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^(2/3)*b^(1/3))))/(3*b)))/(6*b) + (-1/3* 
Cosh[c + d*x]/(b*x*(a + b*x^3)) + (d*((CoshIntegral[d*x]*Sinh[c])/a - (Cos 
hIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(3*a) 
- (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) - (CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/ 
3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a) + (Cosh[c]*Sinh 
Integral[d*x])/a + (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) - (Cosh[c - (a^(1/3)*d)/b^(1/ 
3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a) - (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]...
 

Defintions of rubi rules used

rule 2009

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

rule 5804

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> In 
t[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d 
}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
 

rule 5811

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_ 
)], x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Sinh[c + d*x]/(b*n*(p + 1))) 
, x] - Simp[d*(e^m/(b*n*(p + 1)))   Int[(a + b*x^n)^(p + 1)*Cosh[c + d*x], 
x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 
0] && LtQ[p, -1] && (IntegerQ[n] || GtQ[e, 0])
 

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 = 1.76 (sec) , antiderivative size = 4708, normalized size of antiderivative = 4.26

method	result	size
risch	\(\text {Expression too large to display}\)	\(4708\)

Input:

int(x^4*cosh(d*x+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/108/d^2*(-sum((4*_R1^2*a*b*c*d^3-_R1^2*b^2*c^4-2*_R1*a*b*c^2*d^3+2*_R1* 
b^2*c^5-a^2*d^6+2*a*b*c^3*d^3-b^2*c^6-2*_R1^2*a*b*d^3-16*_R1^2*b^2*c^3+4*_ 
R1*a*b*c*d^3+26*_R1*b^2*c^4+10*a*b*c^2*d^3-10*b^2*c^5-2*_R1*a*b*d^3-16*_R1 
*b^2*c^3-6*a*b*c*d^3+6*b^2*c^4)/(_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^2-sum((4*_R1^2 
*a*b*c*d^3-_R1^2*b^2*c^4-2*_R1*a*b*c^2*d^3+2*_R1*b^2*c^5-a^2*d^6+2*a*b*c^3 
*d^3-b^2*c^6+2*_R1^2*a*b*d^3+16*_R1^2*b^2*c^3-4*_R1*a*b*c*d^3-26*_R1*b^2*c 
^4-10*a*b*c^2*d^3+10*b^2*c^5-2*_R1*a*b*d^3-16*_R1*b^2*c^3-6*a*b*c*d^3+6*b^ 
2*c^4)/(_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^2-sum((4*_R1^2*a*b*c*d^3-_R1^2*b^2*c^4- 
2*_R1*a*b*c^2*d^3+2*_R1*b^2*c^5-a^2*d^6+2*a*b*c^3*d^3-b^2*c^6-2*_R1^2*a*b* 
d^3-16*_R1^2*b^2*c^3+4*_R1*a*b*c*d^3+26*_R1*b^2*c^4+10*a*b*c^2*d^3-10*b^2* 
c^5-2*_R1*a*b*d^3-16*_R1*b^2*c^3-6*a*b*c*d^3+6*b^2*c^4)/(_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^2*x^6+3*exp(d*x+c)*a^2*b^2*d^3*x^3-sum((4*_R1^2*a*b*c*d^3-_R1^2* 
b^2*c^4-2*_R1*a*b*c^2*d^3+2*_R1*b^2*c^5-a^2*d^6+2*a*b*c^3*d^3-b^2*c^6+2*_R 
1^2*a*b*d^3+16*_R1^2*b^2*c^3-4*_R1*a*b*c*d^3-26*_R1*b^2*c^4-10*a*b*c^2*d^3 
+10*b^2*c^5-2*_R1*a*b*d^3-16*_R1*b^2*c^3-6*a*b*c*d^3+6*b^2*c^4)/(_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^2*x^6+sum((_R1^2-2*_R1*c+c^2-6*_R1+6*c+10)/(_R1^2-2*_...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4691 vs. \(2 (805) = 1610\).

Time = 0.17 (sec) , antiderivative size = 4691, normalized size of antiderivative = 4.25 \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^4*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**4*cosh(d*x+c)/(b*x**3+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x^{4} \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \] Input:

integrate(x^4*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="maxima")
 

Output:

1/2*((d^3*x^4*e^(2*c) + 5*d^2*x^3*e^(2*c) + 30*d*x^2*e^(2*c) + 210*x*e^(2* 
c))*e^(d*x) - (d^3*x^4 - 5*d^2*x^3 + 30*d*x^2 - 210*x)*e^(-d*x))/(b^3*d^4* 
x^9*e^c + 3*a*b^2*d^4*x^6*e^c + 3*a^2*b*d^4*x^3*e^c + a^3*d^4*e^c) - 1/2*i 
ntegrate(3*(15*a*d^2*x^2*e^c + (3*a*d^3*e^c - 560*b*e^c)*x^3 + 90*a*d*x*e^ 
c + 70*a*e^c)*e^(d*x)/(b^4*d^4*x^12 + 4*a*b^3*d^4*x^9 + 6*a^2*b^2*d^4*x^6 
+ 4*a^3*b*d^4*x^3 + a^4*d^4), x) + 1/2*integrate(-3*(15*a*d^2*x^2 - (3*a*d 
^3 + 560*b)*x^3 - 90*a*d*x + 70*a)*e^(-d*x)/(b^4*d^4*x^12*e^c + 4*a*b^3*d^ 
4*x^9*e^c + 6*a^2*b^2*d^4*x^6*e^c + 4*a^3*b*d^4*x^3*e^c + a^4*d^4*e^c), x)
 

Giac [F]

\[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x^{4} \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \] Input:

integrate(x^4*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="giac")
 

Output:

integrate(x^4*cosh(d*x + c)/(b*x^3 + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \] Input:

int((x^4*cosh(c + d*x))/(a + b*x^3)^3,x)
 

Output:

int((x^4*cosh(c + d*x))/(a + b*x^3)^3, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\frac {e^{2 c} \left (\int \frac {e^{d x} x^{4}}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right )+\int \frac {x^{4}}{e^{d x} a^{3}+3 e^{d x} a^{2} b \,x^{3}+3 e^{d x} a \,b^{2} x^{6}+e^{d x} b^{3} x^{9}}d x}{2 e^{c}} \] Input:

int(x^4*cosh(d*x+c)/(b*x^3+a)^3,x)
 

Output:

(e**(2*c)*int((e**(d*x)*x**4)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3 
*x**9),x) + int(x**4/(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)