\(\int \frac {x^2 \cosh (c+d x)}{(a+b x^3)^3} \, dx\) [110]

Optimal result

Integrand size = 19, antiderivative size = 781 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.31 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.54 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=-\frac {d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {2 \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-2 \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-2 \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+2 \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}-d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}-d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]+d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-2 \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-2 \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-2 \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))-2 \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}+d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}+d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]-\frac {6 b \cosh (d x) \left (-3 a \cosh (c)+d x \left (a+b x^3\right ) \sinh (c)\right )}{\left (a+b x^3\right )^2}-\frac {6 b \left (d x \left (a+b x^3\right ) \cosh (c)-3 a \sinh (c)\right ) \sinh (d x)}{\left (a+b x^3\right )^2}}{108 a b^2} \] Input:

Integrate[(x^2*Cosh[c + d*x])/(a + b*x^3)^3,x]
 

Output:

-1/108*(d*RootSum[a + b*#1^3 & , (2*Cosh[c + d*#1]*CoshIntegral[d*(x - #1) 
] - 2*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] - 2*Cosh[c + d*#1]*SinhInteg 
ral[d*(x - #1)] + 2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + d*Cosh[c + d 
*#1]*CoshIntegral[d*(x - #1)]*#1 - d*CoshIntegral[d*(x - #1)]*Sinh[c + d*# 
1]*#1 - d*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 + d*Sinh[c + d*#1]*Si 
nhIntegral[d*(x - #1)]*#1)/#1^2 & ] + d*RootSum[a + b*#1^3 & , (-2*Cosh[c 
+ d*#1]*CoshIntegral[d*(x - #1)] - 2*CoshIntegral[d*(x - #1)]*Sinh[c + d*# 
1] - 2*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] - 2*Sinh[c + d*#1]*SinhInte 
gral[d*(x - #1)] + d*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]*#1 + d*CoshIn 
tegral[d*(x - #1)]*Sinh[c + d*#1]*#1 + d*Cosh[c + d*#1]*SinhIntegral[d*(x 
- #1)]*#1 + d*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1)/#1^2 & ] - (6*b* 
Cosh[d*x]*(-3*a*Cosh[c] + d*x*(a + b*x^3)*Sinh[c]))/(a + b*x^3)^2 - (6*b*( 
d*x*(a + b*x^3)*Cosh[c] - 3*a*Sinh[c])*Sinh[d*x])/(a + b*x^3)^2)/(a*b^2)
 

Rubi [A] (verified)

Time = 2.10 (sec) , antiderivative size = 863, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5812, 5801, 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.

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

\(\Big \downarrow \) 5812

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

\(\Big \downarrow \) 5801

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

\(\Big \downarrow \) 5815

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^2*Cosh[c + d*x])/(a + b*x^3)^3,x]
 

Output:

-1/6*Cosh[c + d*x]/(b*(a + b*x^3)^2) + (d*(-1/3*Sinh[c + d*x]/(b*x^2*(a + 
b*x^3)) - (2*(-1/2*(d*Cosh[c + d*x])/(a*x) + (d^2*CoshIntegral[d*x]*Sinh[c 
])/(2*a) - (b^(2/3)*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1 
/3)*d)/b^(1/3)])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*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 
^(5/3)) - ((-1)^(2/3)*b^(2/3)*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^(5/3)) - Sinh[c + 
 d*x]/(2*a*x^2) + (d^2*Cosh[c]*SinhIntegral[d*x])/(2*a) - ((-1)^(1/3)*b^(2 
/3)*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^(5/3)) - (b^(2/3)*Cosh[c - (a^(1/3)*d)/b^(1/3) 
]*SinhIntegral[(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)]*SinhIntegral[((-1)^(2/3)*a^(1/ 
3)*d)/b^(1/3) + d*x])/(3*a^(5/3))))/(3*b) + (d*(-(Cosh[c + d*x]/(a*x)) + ( 
(-1)^(2/3)*b^(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])/(3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*Co 
sh[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^(4/3)) + (b^(1/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*C 
oshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(4/3)) + (d*CoshIntegral[d*x] 
*Sinh[c])/a + (d*Cosh[c]*SinhIntegral[d*x])/a - ((-1)^(2/3)*b^(1/3)*Sinh[c 
 + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/...
 

Defintions of rubi rules used

rule 2009

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

rule 5801

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Si 
mp[x^(-n + 1)*(a + b*x^n)^(p + 1)*(Sinh[c + d*x]/(b*n*(p + 1))), x] + (-Sim 
p[(-n + 1)/(b*n*(p + 1))   Int[((a + b*x^n)^(p + 1)*Sinh[c + d*x])/x^n, x], 
 x] - Simp[d/(b*n*(p + 1))   Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d* 
x], x], x]) /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p 
, -1] && GtQ[n, 2]
 

rule 5812

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p 
_), x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Cosh[c + d*x]/(b*n*(p + 1))) 
, x] - Simp[d*(e^m/(b*n*(p + 1)))   Int[(a + b*x^n)^(p + 1)*Sinh[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 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 = 1.53 (sec) , antiderivative size = 2238, normalized size of antiderivative = 2.87

method	result	size
risch	\(\text {Expression too large to display}\)	\(2238\)

Input:

int(x^2*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^2*x^6-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^2*x^6+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=Root 
Of(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))*b^2*c*x^6+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))*b^2*c*x^6+3*a*b^2*d*x^4*exp(d*x+c)-sum((_R2^2*b*c^2-_R2* 
a*d^3-2*_R2*b*c^3-a*c*d^3+b*c^4-8*_R2^2*b*c+10*_R2*b*c^2+2*a*d^3-2*b*c^3+8 
*_R2*b*c+2*b*c^2)/(_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))*b^2*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^2*x^3-3*a*b^2*d*x^4*exp 
(-d*x-c)-sum((_R2^2*b*c^2-_R2*a*d^3-2*_R2*b*c^3-a*c*d^3+b*c^4+8*_R2^2*b*c- 
10*_R2*b*c^2-2*a*d^3+2*b*c^3+8*_R2*b*c+2*b*c^2)/(_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))* 
b^2*x^6-2*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)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2972 vs. \(2 (559) = 1118\).

Time = 0.16 (sec) , antiderivative size = 2972, normalized size of antiderivative = 3.81 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^2*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="fricas")
 

Output:

-1/216*(36*a^2*cosh(d*x + c) + ((a*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^ 
2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a 
*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2) + 2* 
(a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^ 
3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 
 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2))*Ei(d*x - 1/2*(a*d^3/b)^(1/3)*(sqrt( 
-3) + 1))*cosh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1) + c) + ((-a*d^3/b)^(2/3) 
*((b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh( 
d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + 
a^2))*sinh(d*x + c)^2) + 2*(-a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + 
sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x 
^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2))*Ei(-d*x 
 - 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3 
) + 1) - c) + ((a*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2 
*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 + sq 
rt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2) + 2*(a*d^3/b)^(1/3)*( 
(b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d* 
x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^ 
2))*sinh(d*x + c)^2))*Ei(d*x + 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1))*cosh(1/ 
2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1) - c) + ((-a*d^3/b)^(2/3)*((b^2*x^6 + 2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**2*cosh(d*x+c)/(b*x**3+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="maxima")
 

Output:

1/2*((d*x^2*e^(2*c) + 7*x*e^(2*c))*e^(d*x) - (d*x^2 - 7*x)*e^(-d*x))/(b^3* 
d^2*x^9*e^c + 3*a*b^2*d^2*x^6*e^c + 3*a^2*b*d^2*x^3*e^c + a^3*d^2*e^c) + 1 
/2*integrate((56*b*x^3*e^c - 9*a*d*x*e^c - 7*a*e^c)*e^(d*x)/(b^4*d^2*x^12 
+ 4*a*b^3*d^2*x^9 + 6*a^2*b^2*d^2*x^6 + 4*a^3*b*d^2*x^3 + a^4*d^2), x) + 1 
/2*integrate((56*b*x^3 + 9*a*d*x - 7*a)*e^(-d*x)/(b^4*d^2*x^12*e^c + 4*a*b 
^3*d^2*x^9*e^c + 6*a^2*b^2*d^2*x^6*e^c + 4*a^3*b*d^2*x^3*e^c + a^4*d^2*e^c 
), x)
 

Giac [F]

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

integrate(x^2*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="giac")
 

Output:

integrate(x^2*cosh(d*x + c)/(b*x^3 + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \] Input:

int((x^2*cosh(c + d*x))/(a + b*x^3)^3,x)
 

Output:

int((x^2*cosh(c + d*x))/(a + b*x^3)^3, x)
 

Reduce [F]

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

int(x^2*cosh(d*x+c)/(b*x^3+a)^3,x)
 

Output:

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