\(\int \frac {\cosh (c+d x)}{(a+b x^3)^2} \, dx\) [105]

Optimal result

Integrand size = 16, antiderivative size = 739 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.14 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.52 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\frac {6 b x \cosh (c+d x)+\left (a+b x^3\right ) \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {2 \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-2 \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-2 \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+2 \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}-d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}-d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]-\left (a+b x^3\right ) \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-2 \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-2 \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-2 \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))-2 \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}+d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}+d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]}{18 a b \left (a+b x^3\right )} \] Input:

Integrate[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 & , (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]*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 & ] - (a + b*x^3) 
*RootSum[a + b*#1^3 & , (-2*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] - 2*Co 
shIntegral[d*(x - #1)]*Sinh[c + d*#1] - 2*Cosh[c + d*#1]*SinhIntegral[d*(x 
 - #1)] - 2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + d*Cosh[c + d*#1]*Cos 
hIntegral[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]*SinhIntegr 
al[d*(x - #1)]*#1)/#1^2 & ])/(18*a*b*(a + b*x^3))
 

Rubi [A] (verified)

Time = 2.32 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5802, 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.

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

\(\Big \downarrow \) 5802

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

\(\Big \downarrow \) 5815

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/3*Cosh[c + d*x]/(b*x^2*(a + b*x^3)) + (d*((d*Cosh[c]*CoshIntegral[d*x]) 
/a + (b^(1/3)*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d) 
/b^(1/3)])/(3*a^(4/3)) + ((-1)^(2/3)*b^(1/3)*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^(4/3) 
) - ((-1)^(1/3)*b^(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^(4/3)) - Sinh[c + d*x]/ 
(a*x) + (d*Sinh[c]*SinhIntegral[d*x])/a - ((-1)^(2/3)*b^(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^(4/3)) + (b^(1/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[( 
a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(4/3)) - ((-1)^(1/3)*b^(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^(4/3))))/(3*b) - (2*(-1/2*Cosh[c + d*x]/(a*x^2) + (d^2*Cosh[c]* 
CoshIntegral[d*x])/(2*a) + ((-1)^(1/3)*b^(2/3)*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^(5/ 
3)) - ((-1)^(2/3)*b^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshInt 
egral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a^(5/3)) - (b^(2/3)*Cos 
h[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^( 
5/3)) - (d*Sinh[c + d*x])/(2*a*x) + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a) 
- ((-1)^(1/3)*b^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegra 
l[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(5/3)) - (b^(2/3)*Sinh[c ...
 

Defintions of rubi rules used

rule 2009

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

rule 5802

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Si 
mp[x^(-n + 1)*(a + b*x^n)^(p + 1)*(Cosh[c + d*x]/(b*n*(p + 1))), x] + (-Sim 
p[(-n + 1)/(b*n*(p + 1))   Int[((a + b*x^n)^(p + 1)*Cosh[c + d*x])/x^n, x], 
 x] - Simp[d/(b*n*(p + 1))   Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d* 
x], x], x]) /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p 
, -1] && 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 = 0.76 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.31

method	result	size
risch	\(\frac {d^{3} {\mathrm e}^{-d x -c} x}{6 a \left (b \,d^{3} x^{3}+d^{3} a \right )}-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1} -c +2\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{18 a b}+\frac {d^{3} {\mathrm e}^{d x +c} x}{6 a \left (b \,d^{3} x^{3}+d^{3} a \right )}+\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1} -c -2\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{18 a b}\)	\(226\)

Input:

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

Output:

1/6*d^3*exp(-d*x-c)*x/a/(b*d^3*x^3+a*d^3)-1/18*d^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))+1/6*d^3*exp(d*x+c)*x/a/(b*d^3*x^3+a*d^3)+1/18*d^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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2048 vs. \(2 (519) = 1038\).

Time = 0.14 (sec) , antiderivative size = 2048, normalized size of antiderivative = 2.77 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(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) + 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) + 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) + 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 
) + 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(...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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