\(\int \frac {x^3 \cosh (c+d x)}{a+b x^3} \, dx\) [95]

Optimal result

Integrand size = 19, antiderivative size = 358 \[ \int \frac {x^3 \cosh (c+d x)}{a+b x^3} \, dx=\frac {\sqrt [3]{-1} \sqrt [3]{a} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac {\sqrt [3]{a} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}}+\frac {\sinh (c+d x)}{b d}-\frac {\sqrt [3]{-1} \sqrt [3]{a} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac {\sqrt [3]{a} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.55 \[ \int \frac {x^3 \cosh (c+d x)}{a+b x^3} \, dx=-\frac {a d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]+a d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]-6 b \sinh (c+d x)}{6 b^2 d} \] Input:

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

Output:

-1/6*(a*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]*SinhIntegral[d* 
(x - #1)] + Sinh[c + d*#1]*SinhIntegral[d*(x - #1)])/#1^2 & ] + a*d*RootSu 
m[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)])/#1^2 & ] - 6*b*Sinh[c + d*x])/(b^2*d)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {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.

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((-1)^(1/3)*a^(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*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*C 
osh[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*b^(4/3)) - (a^(1/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]* 
CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3)) + Sinh[c + d*x]/(b*d) 
 - ((-1)^(1/3)*a^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegr 
al[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(4/3)) - (a^(1/3)*Sinh[c - 
(a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3)) 
- ((-1)^(2/3)*a^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegra 
l[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3))
 

Defintions of rubi rules used

rule 2009

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

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 = 0.60 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.87

method	result	size
risch	\(\frac {c^{3} \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 {{\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 d b}+\frac {c^{3} \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 {{\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 d b}-\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 {\textit {\_R1} \,{\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{2 d b}-\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 {\textit {\_R1} \,{\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{2 d 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 {\textit {\_R1}^{2} {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{2 d 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 {\textit {\_R1}^{2} {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{2 d b}+\frac {{\mathrm e}^{d x +c}}{2 b d}-\frac {{\mathrm e}^{-d x -c}}{2 d b}-\frac {\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 (3 \textit {\_R1}^{2} b c -3 \textit {\_R1} b \,c^{2}-d^{3} a +b \,c^{3}\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{6 d \,b^{2}}-\frac {\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 (3 \textit {\_R1}^{2} b c -3 \textit {\_R1} b \,c^{2}-d^{3} a +b \,c^{3}\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{6 d \,b^{2}}\)	\(671\)

Input:

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

Output:

1/6/d/b*c^3*sum(1/(_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/b*c^3*sum(1/(_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))-1/2/d/b*c^2*sum(_R1/(_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))-1/2/d/b*c^2*su 
m(_R1/(_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))+1/2/d/b*c*sum(_R1^2/(_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))+1/2/d/b*c*sum(_R1^2/(_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))+1/2/b/d*exp(d*x+c)-1/2/d 
/b*exp(-d*x-c)-1/6/d/b^2*sum((3*_R1^2*b*c-3*_R1*b*c^2-a*d^3+b*c^3)/(_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/b^2*sum((3*_R1^2*b*c-3*_R1*b*c^2-a*d^3+b*c^3)/(_R 
1^2-2*_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. 977 vs. \(2 (250) = 500\).

Time = 0.12 (sec) , antiderivative size = 977, normalized size of antiderivative = 2.73 \[ \int \frac {x^3 \cosh (c+d x)}{a+b x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*((a*d^3/b)^(1/3)*((sqrt(-3) + 1)*cosh(d*x + c)^2 - (sqrt(-3) + 1)*sin 
h(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)*((sqrt(-3) + 1)*cosh(d*x 
+ c)^2 - (sqrt(-3) + 1)*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)*((sqrt(-3) - 1)*cosh(d*x + c)^2 - (sqrt(-3) - 1)*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)*((sqrt(-3) - 1)*cosh(d*x + c)^2 - (sqrt(-3) 
 - 1)*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)*(cosh(d*x + 
 c)^2 - 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)*(cosh(d*x + c)^2 - 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)*((sqrt(-3) + 1) 
*cosh(d*x + c)^2 - (sqrt(-3) + 1)*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)*((sqrt(-3) + 1)*cosh(d*x + c)^2 - (sqrt(-3) + 1)*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)*((sqrt(-3) - 1)*cosh(d*x + c)^2 - 
 (sqrt(-3) - 1)*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)*((s...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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