\(\int \frac {x^2 \cosh (c+d x)}{(a+b x^3)^2} \, dx\) [103]

Optimal result

Integrand size = 19, antiderivative size = 373 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=-\frac {\cosh (c+d x)}{3 b \left (a+b x^3\right )}+\frac {d \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac {\sqrt [3]{-1} d \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac {\sqrt [3]{-1} d \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac {d \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.12 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.54 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\frac {-\frac {6 b \cosh (c+d x)}{a+b x^3}-d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]+d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{18 b^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5812, 5803, 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.

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

\(\Big \downarrow \) 5812

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

\(\Big \downarrow \) 5803

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 5803

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> In 
t[ExpandIntegrand[Sinh[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 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])
 

Maple [C] (warning: unable to verify)

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

Time = 0.87 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.59

method	result	size
risch	\(-\frac {d^{3} {\mathrm e}^{-d x -c}}{6 b \left (b \,d^{3} x^{3}+d^{3} a \right )}-\frac {\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (2 \textit {\_R2}^{2} b c -3 \textit {\_R2} b \,c^{2}-d^{3} a +b \,c^{3}+2 \textit {\_R2} b c \right ) {\mathrm e}^{-\textit {\_R2}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R2} +c \right )}{\textit {\_R2}^{2}-2 \textit {\_R2} c +c^{2}}}{18 a \,b^{2}}-\frac {c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1} -c +2\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{18 a b}+\frac {c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-\textit {\_R1} c +\textit {\_R1} +c \right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a b}-\frac {d^{3} {\mathrm e}^{d x +c}}{6 b \left (b \,d^{3} x^{3}+d^{3} a \right )}+\frac {\munderset {\textit {\_R2} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (2 \textit {\_R2}^{2} b c -3 \textit {\_R2} b \,c^{2}-d^{3} a +b \,c^{3}-2 \textit {\_R2} b c \right ) {\mathrm e}^{\textit {\_R2}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R2} -c \right )}{\textit {\_R2}^{2}-2 \textit {\_R2} c +c^{2}}}{18 a \,b^{2}}+\frac {c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1} -c -2\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{18 a b}-\frac {c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-\textit {\_R1} c -\textit {\_R1} -c \right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a b}\)	\(594\)

Input:

int(x^2*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)-1/18/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))-1/18*c^2/a/b*sum(( 
_R1-c+2)/(_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))+1/9*c/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^3*exp(d*x+c)/b/(b*d^3*x^3+a*d^3)+1/18/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)) 
+1/18*c^2/a/b*sum((_R1-c-2)/(_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))-1/9*c/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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1276 vs. \(2 (263) = 526\).

Time = 0.14 (sec) , antiderivative size = 1276, normalized size of antiderivative = 3.42 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*((d*x^2*e^(2*c) + 4*x*e^(2*c))*e^(d*x) - (d*x^2 - 4*x)*e^(-d*x))/(b^2* 
d^2*x^6*e^c + 2*a*b*d^2*x^3*e^c + a^2*d^2*e^c) + 1/2*integrate(2*(10*b*x^3 
*e^c - 3*a*d*x*e^c - 2*a*e^c)*e^(d*x)/(b^3*d^2*x^9 + 3*a*b^2*d^2*x^6 + 3*a 
^2*b*d^2*x^3 + a^3*d^2), x) + 1/2*integrate(2*(10*b*x^3 + 3*a*d*x - 2*a)*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), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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