\(\int \frac {x^3 \cosh (c+d x)}{(a+b x^3)^2} \, dx\) [102]

Optimal result

Integrand size = 19, antiderivative size = 718 \[ \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Output:

-1/3*x*cosh(d*x+c)/b/(b*x^3+a)-1/9*(-1)^(1/3)*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^(2/3)/b^(4/3)+1/9*(-1)^(2 
/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^(2/3)/b^(4/3)+1/9*cosh(c-a^(1/3)*d/b^(1/3))*Chi(a^(1/3)*d/b^(1/3)+ 
d*x)/a^(2/3)/b^(4/3)-1/9*d*Chi(a^(1/3)*d/b^(1/3)+d*x)*sinh(c-a^(1/3)*d/b^( 
1/3))/a^(1/3)/b^(5/3)-1/9*(-1)^(2/3)*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^(1/3)/b^(5/3)+1/9*(-1)^(1/3)*d*C 
hi(-(-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^(1/3)/b^(5/3)-1/9*(-1)^(2/3)*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^(1/3)/b^(5/3)-1/9*(-1)^(1/3)*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^(2/ 
3)/b^(4/3)-1/9*d*cosh(c-a^(1/3)*d/b^(1/3))*Shi(a^(1/3)*d/b^(1/3)+d*x)/a^(1 
/3)/b^(5/3)+1/9*sinh(c-a^(1/3)*d/b^(1/3))*Shi(a^(1/3)*d/b^(1/3)+d*x)/a^(2/ 
3)/b^(4/3)+1/9*(-1)^(1/3)*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^(1/3)/b^(5/3)+1/9*(-1)^(2/3)*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^(2/3)/b^(4/ 
3)
 

Mathematica [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.51 \[ \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\frac {-\frac {6 b x \cosh (c+d x)}{a+b x^3}-\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))-\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}-d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}-d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}+d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}+d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]}{18 b^2} \] Input:

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

Output:

((-6*b*x*Cosh[c + d*x])/(a + b*x^3) - RootSum[a + b*#1^3 & , (-(Cosh[c + d 
*#1]*CoshIntegral[d*(x - #1)]) + CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] + 
 Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] - 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]*SinhIntegral[d*(x - #1)]*#1)/#1^2 & ] + RootSum[a + b 
*#1^3 & , (Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] + CoshIntegral[d*(x - # 
1)]*Sinh[c + d*#1] + Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] + 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]*SinhIn 
tegral[d*(x - #1)]*#1 + d*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1)/#1^2 
 & ])/(18*b^2)
 

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5814, 5804, 2009, 5815, 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.

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

\(\Big \downarrow \) 5814

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

\(\Big \downarrow \) 5804

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5815

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 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])
 

Maple [C] (warning: unable to verify)

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

Time = 0.98 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.22

method	result	size
risch	\(\text {Expression too large to display}\)	\(877\)

Input:

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

Output:

-1/6*d^3*exp(-d*x-c)/b/(b*d^3*x^3+a*d^3)*x-1/18/d/a/b^2*sum((3*_R2^2*b*c^2 
-_R2*a*d^3-5*_R2*b*c^3-2*a*c*d^3+2*b*c^4+3*_R2*b*c^2+a*d^3-b*c^3)/(_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))+1/18/d*c^3/a/b*sum((_R1-c+2)/(_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))-1 
/6/d*c^2/a/b*sum((_R1^2-_R1*c+_R1+c)/(_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))+1/6/d*c/a/b 
^2*sum((2*_R2^2*b*c-3*_R2*b*c^2-a*d^3+b*c^3+2*_R2*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))-1/6*d^3*exp(d*x+c)/b/(b*d^3*x^3+a*d^3)*x+1/18/d/a/b^2*sum((3*_R2^2* 
b*c^2-_R2*a*d^3-5*_R2*b*c^3-2*a*c*d^3+2*b*c^4-3*_R2*b*c^2-a*d^3+b*c^3)/(_R 
2^2-2*_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))-1/18/d*c^3/a/b*sum((_R1-c-2)/(_R1^2-2*_R1*c+c^2)*ex 
p(_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))+1/6/d*c^2/a/b*sum((_R1^2-_R1*c-_R1-c)/(_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))-1/6/d* 
c/a/b^2*sum((2*_R2^2*b*c-3*_R2*b*c^2-a*d^3+b*c^3-2*_R2*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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2050 vs. \(2 (500) = 1000\).

Time = 0.12 (sec) , antiderivative size = 2050, normalized size of antiderivative = 2.86 \[ \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/36*(12*a*d*x*cosh(d*x + c) - ((a*d^3/b)^(2/3)*((b*x^3 - sqrt(-3)*(b*x^3 
 + a) + a)*cosh(d*x + c)^2 - (b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*sinh(d*x + 
 c)^2) - (a*d^3/b)^(1/3)*((b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c) 
^2 - (b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*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*x^3 - sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - 
(b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*sinh(d*x + c)^2) - (-a*d^3/b)^(1/3)*((b 
*x^3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)*(b*x^ 
3 + a) + a)*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*x^ 
3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)*(b*x^3 + 
 a) + a)*sinh(d*x + c)^2) - (a*d^3/b)^(1/3)*((b*x^3 - sqrt(-3)*(b*x^3 + a) 
 + a)*cosh(d*x + c)^2 - (b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*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*x^3 + sqrt(-3)*(b*x^3 + a) + a) 
*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*sinh(d*x + c)^2) - ( 
-a*d^3/b)^(1/3)*((b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x 
^3 - sqrt(-3)*(b*x^3 + a) + a)*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) - 2*(( 
-a*d^3/b)^(2/3)*((b*x^3 + a)*cosh(d*x + c)^2 - (b*x^3 + a)*sinh(d*x + c...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*((d^2*x^3*e^(2*c) + 3*d*x^2*e^(2*c) + 12*x*e^(2*c))*e^(d*x) - (d^2*x^3 
 - 3*d*x^2 + 12*x)*e^(-d*x))/(b^2*d^3*x^6*e^c + 2*a*b*d^3*x^3*e^c + a^2*d^ 
3*e^c) + 1/2*integrate(-6*(a*d^2*x^2*e^c - 10*b*x^3*e^c + 3*a*d*x*e^c + 2* 
a*e^c)*e^(d*x)/(b^3*d^3*x^9 + 3*a*b^2*d^3*x^6 + 3*a^2*b*d^3*x^3 + a^3*d^3) 
, x) - 1/2*integrate(-6*(a*d^2*x^2 - 10*b*x^3 - 3*a*d*x + 2*a)*e^(-d*x)/(b 
^3*d^3*x^9*e^c + 3*a*b^2*d^3*x^6*e^c + 3*a^2*b*d^3*x^3*e^c + a^3*d^3*e^c), 
 x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\frac {-e^{2 d x +2 c} d x +e^{d x +2 c} \left (\int \frac {e^{d x}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} d +e^{d x +2 c} \left (\int \frac {e^{d x}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a b d \,x^{3}+e^{d x +2 c} \left (\int \frac {e^{d x} x^{4}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a b \,d^{2}+e^{d x +2 c} \left (\int \frac {e^{d x} x^{4}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{2} d^{2} x^{3}+e^{d x +2 c} \left (\int \frac {e^{d x} x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} d^{2}+e^{d x +2 c} \left (\int \frac {e^{d x} x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a b \,d^{2} x^{3}-e^{d x} \left (\int \frac {x^{4}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{3}+e^{d x} b^{2} x^{6}}d x \right ) a b \,d^{2}-e^{d x} \left (\int \frac {x^{4}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{3}+e^{d x} b^{2} x^{6}}d x \right ) b^{2} d^{2} x^{3}-e^{d x} \left (\int \frac {x^{3}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{3}+e^{d x} b^{2} x^{6}}d x \right ) a b d -e^{d x} \left (\int \frac {x^{3}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{3}+e^{d x} b^{2} x^{6}}d x \right ) b^{2} d \,x^{3}-3 e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{3}+e^{d x} b^{2} x^{6}}d x \right ) a b -3 e^{d x} \left (\int \frac {x^{2}}{e^{d x} a^{2}+2 e^{d x} a b \,x^{3}+e^{d x} b^{2} x^{6}}d x \right ) b^{2} x^{3}-e^{d x} \left (\int \frac {x}{e^{d x} a^{2}+2 e^{d x} a b \,x^{3}+e^{d x} b^{2} x^{6}}d x \right ) a^{2} d^{2}-e^{d x} \left (\int \frac {x}{e^{d x} a^{2}+2 e^{d x} a b \,x^{3}+e^{d x} b^{2} x^{6}}d x \right ) a b \,d^{2} x^{3}-d x -1}{4 e^{d x +c} b d \left (b \,x^{3}+a \right )} \] Input:

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

Output:

( - e**(2*c + 2*d*x)*d*x + e**(2*c + d*x)*int(e**(d*x)/(a**2 + 2*a*b*x**3 
+ b**2*x**6),x)*a**2*d + e**(2*c + d*x)*int(e**(d*x)/(a**2 + 2*a*b*x**3 + 
b**2*x**6),x)*a*b*d*x**3 + e**(2*c + d*x)*int((e**(d*x)*x**4)/(a**2 + 2*a* 
b*x**3 + b**2*x**6),x)*a*b*d**2 + e**(2*c + d*x)*int((e**(d*x)*x**4)/(a**2 
 + 2*a*b*x**3 + b**2*x**6),x)*b**2*d**2*x**3 + e**(2*c + d*x)*int((e**(d*x 
)*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x)*a**2*d**2 + e**(2*c + d*x)*int((e* 
*(d*x)*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x)*a*b*d**2*x**3 - e**(d*x)*int( 
x**4/(e**(d*x)*a**2 + 2*e**(d*x)*a*b*x**3 + e**(d*x)*b**2*x**6),x)*a*b*d** 
2 - e**(d*x)*int(x**4/(e**(d*x)*a**2 + 2*e**(d*x)*a*b*x**3 + e**(d*x)*b**2 
*x**6),x)*b**2*d**2*x**3 - e**(d*x)*int(x**3/(e**(d*x)*a**2 + 2*e**(d*x)*a 
*b*x**3 + e**(d*x)*b**2*x**6),x)*a*b*d - e**(d*x)*int(x**3/(e**(d*x)*a**2 
+ 2*e**(d*x)*a*b*x**3 + e**(d*x)*b**2*x**6),x)*b**2*d*x**3 - 3*e**(d*x)*in 
t(x**2/(e**(d*x)*a**2 + 2*e**(d*x)*a*b*x**3 + e**(d*x)*b**2*x**6),x)*a*b - 
 3*e**(d*x)*int(x**2/(e**(d*x)*a**2 + 2*e**(d*x)*a*b*x**3 + e**(d*x)*b**2* 
x**6),x)*b**2*x**3 - e**(d*x)*int(x/(e**(d*x)*a**2 + 2*e**(d*x)*a*b*x**3 + 
 e**(d*x)*b**2*x**6),x)*a**2*d**2 - e**(d*x)*int(x/(e**(d*x)*a**2 + 2*e**( 
d*x)*a*b*x**3 + e**(d*x)*b**2*x**6),x)*a*b*d**2*x**3 - d*x - 1)/(4*e**(c + 
 d*x)*b*d*(a + b*x**3))