\(\int \frac {\cosh (c+d x)}{x (a+b x^3)^2} \, dx\) [106]

Optimal result

Integrand size = 19, antiderivative size = 697 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.36 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.59 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\frac {\frac {6 a \cosh (c) \cosh (d x)}{a+b x^3}+18 \cosh (c) \text {Chi}(d x)-3 \text {RootSum}\left [a+b \text {$\#$1}^3\&,\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))\&\right ]-3 \text {RootSum}\left [a+b \text {$\#$1}^3\&,\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))\&\right ]+\frac {a d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{b}-\frac {a d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{b}+\frac {6 a \sinh (c) \sinh (d x)}{a+b x^3}+18 \sinh (c) \text {Shi}(d x)}{18 a^2} \] Input:

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

Output:

((6*a*Cosh[c]*Cosh[d*x])/(a + b*x^3) + 18*Cosh[c]*CoshIntegral[d*x] - 3*Ro 
otSum[a + b*#1^3 & , Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] - CoshIntegra 
l[d*(x - #1)]*Sinh[c + d*#1] - Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] + S 
inh[c + d*#1]*SinhIntegral[d*(x - #1)] & ] - 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]*SinhIntegra 
l[d*(x - #1)] & ] + (a*d*RootSum[a + b*#1^3 & , (Cosh[c + d*#1]*CoshIntegr 
al[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 & 
 ])/b - (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 & ])/b + (6*a 
*Sinh[c]*Sinh[d*x])/(a + b*x^3) + 18*Sinh[c]*SinhIntegral[d*x])/(18*a^2)
 

Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 846, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {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.

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

\(\Big \downarrow \) 5814

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

\(\Big \downarrow \) 5815

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\int \frac {\cosh (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}+\frac {d \left (-\frac {b^{2/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {\sqrt [3]{-1} b^{2/3} \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}+\frac {d^2 \sinh (c) \text {Chi}(d x)}{2 a}+\frac {d^2 \cosh (c) \text {Shi}(d x)}{2 a}-\frac {\sinh (c+d x)}{2 a x^2}-\frac {d \cosh (c+d x)}{2 a x}\right )}{3 b}-\frac {\cosh (c+d x)}{3 b x^3 \left (a+b x^3\right )}\)

\(\Big \downarrow \) 5816

\(\displaystyle -\frac {\int \left (\frac {b^2 \cosh (c+d x) x^2}{a^2 \left (b x^3+a\right )}-\frac {b \cosh (c+d x)}{a^2 x}+\frac {\cosh (c+d x)}{a x^4}\right )dx}{b}+\frac {d \left (-\frac {b^{2/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {\sqrt [3]{-1} b^{2/3} \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}+\frac {d^2 \sinh (c) \text {Chi}(d x)}{2 a}+\frac {d^2 \cosh (c) \text {Shi}(d x)}{2 a}-\frac {\sinh (c+d x)}{2 a x^2}-\frac {d \cosh (c+d x)}{2 a x}\right )}{3 b}-\frac {\cosh (c+d x)}{3 b x^3 \left (a+b x^3\right )}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/3*Cosh[c + d*x]/(b*x^3*(a + b*x^3)) + (d*(-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) - 
 (-1/3*Cosh[c + d*x]/(a*x^3) - (d^2*Cosh[c + d*x])/(6*a*x) - (b*Cosh[c]*Co 
shIntegral[d*x])/a^2 + (b*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshInt 
egral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^2) + (b*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) + (b*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/ 
b^(1/3) + d*x])/(3*a^2) + (d^3*CoshIntegral[d*x]*Sinh[c])/(6*a) - (d*Sinh[ 
c + d*x])/(6*a*x^2) + (d^3*Cosh[c]*SinhIntegral[d*x])/(6*a) - (b*Sinh[c]*S 
inhIntegral[d*x])/a^2 - (b*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*Sin...
 

Defintions of rubi rules used

rule 2009

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

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 = 0.88 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.48

method	result	size
risch	\(\frac {{\mathrm e}^{-d x -c} d^{3}}{6 a \left (b \left (d x +c \right )^{3}-3 \left (d x +c \right )^{2} b c +3 \left (d x +c \right ) b \,c^{2}+d^{3} a -b \,c^{3}\right )}-\frac {{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right )}{2 a^{2}}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (-d^{3} a +3 \textit {\_R1}^{2} b -6 \textit {\_R1} b c +3 b \,c^{2}\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{18 a^{2} b}+\frac {{\mathrm e}^{d x +c} d^{3}}{6 a \left (b \left (d x +c \right )^{3}-3 \left (d x +c \right )^{2} b c +3 \left (d x +c \right ) b \,c^{2}+d^{3} a -b \,c^{3}\right )}-\frac {{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right )}{2 a^{2}}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (d^{3} a +3 \textit {\_R1}^{2} b -6 \textit {\_R1} b c +3 b \,c^{2}\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{18 a^{2} b}\)	\(338\)

Input:

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

Output:

1/6*exp(-d*x-c)*d^3/a/(b*(d*x+c)^3-3*(d*x+c)^2*b*c+3*(d*x+c)*b*c^2+d^3*a-b 
*c^3)-1/2/a^2*exp(-c)*Ei(1,d*x)+1/18/a^2/b*sum((-a*d^3+3*_R1^2*b-6*_R1*b*c 
+3*b*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))+1/6*exp(d*x+c)*d^3/a/(b*(d*x+c)^3-3*(d 
*x+c)^2*b*c+3*(d*x+c)*b*c^2+d^3*a-b*c^3)-1/2/a^2*exp(c)*Ei(1,-d*x)+1/18/a^ 
2/b*sum((a*d^3+3*_R1^2*b-6*_R1*b*c+3*b*c^2)/(_R1^2-2*_R1*c+c^2)*exp(_R1)*E 
i(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. 1773 vs. \(2 (509) = 1018\).

Time = 0.14 (sec) , antiderivative size = 1773, normalized size of antiderivative = 2.54 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: AttributeError >> type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\int \frac {\cosh \left (d x +c \right )}{b^{2} x^{7}+2 a b \,x^{4}+a^{2} x}d x \] Input:

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

Output:

int(cosh(c + d*x)/(a**2*x + 2*a*b*x**4 + b**2*x**7),x)