\(\int \frac {\cosh (c+d x)}{x^3 (a+b x^3)} \, dx\) [101]

Optimal result

Integrand size = 19, antiderivative size = 410 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=-\frac {\cosh (c+d x)}{2 a x^2}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {\sqrt [3]{-1} b^{2/3} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{5/3}}-\frac {d \sinh (c+d x)}{2 a x}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {\sqrt [3]{-1} b^{2/3} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{5/3}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.19 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.58 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=-\frac {3 \cosh (c+d x)-3 d^2 x^2 \cosh (c) \text {Chi}(d x)+x^2 \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]+x^2 \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]+3 d x \sinh (c+d x)-3 d^2 x^2 \sinh (c) \text {Shi}(d x)}{6 a x^2} \] Input:

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

Output:

-1/6*(3*Cosh[c + d*x] - 3*d^2*x^2*Cosh[c]*CoshIntegral[d*x] + x^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)])/#1^2 & ] + x^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]*SinhInt 
egral[d*(x - #1)])/#1^2 & ] + 3*d*x*Sinh[c + d*x] - 3*d^2*x^2*Sinh[c]*Sinh 
Integral[d*x])/(a*x^2)
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 410, 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.

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-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)]*CoshIntegral[-(((-1)^(2/3)*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)]*CoshI 
ntegral[(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)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) 
 - d*x])/(3*a^(5/3)) - (b^(2/3)*Sinh[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)*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^(5/3))
 

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 = 0.70 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.55

method	result	size
risch	\(-\frac {3 \,{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right ) d^{2} x^{2}+3 \,{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right ) d^{2} x^{2}-2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right ) d^{2} x^{2}-2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right ) d^{2} x^{2}-3 \,{\mathrm e}^{-d x -c} d x +3 d x \,{\mathrm e}^{d x +c}+3 \,{\mathrm e}^{-d x -c}+3 \,{\mathrm e}^{d x +c}}{12 a \,x^{2}}\)	\(226\)

Input:

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

Output:

-1/12*(3*exp(c)*Ei(1,-d*x)*d^2*x^2+3*exp(-c)*Ei(1,d*x)*d^2*x^2-2*sum(1/(_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))*d^2*x^2-2*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))*d^2*x^2-3 
*exp(-d*x-c)*d*x+3*d*x*exp(d*x+c)+3*exp(-d*x-c)+3*exp(d*x+c))/a/x^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1251 vs. \(2 (294) = 588\).

Time = 0.14 (sec) , antiderivative size = 1251, normalized size of antiderivative = 3.05 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/12*(6*a*d^2*x*sinh(d*x + c) - (a*d^3/b)^(1/3)*((sqrt(-3)*b*x^2 + b*x^2) 
*cosh(d*x + c)^2 - (sqrt(-3)*b*x^2 + b*x^2)*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)*b*x^2 + b*x^2)*cosh(d*x + c)^2 - (sqrt(-3 
)*b*x^2 + b*x^2)*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)*((s 
qrt(-3)*b*x^2 - b*x^2)*cosh(d*x + c)^2 - (sqrt(-3)*b*x^2 - b*x^2)*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)*b*x^2 - b*x^2)*cosh 
(d*x + c)^2 - (sqrt(-3)*b*x^2 - b*x^2)*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*(b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*(-a*d^3/b)^(1/3)*Ei(- 
d*x + (-a*d^3/b)^(1/3))*cosh(c + (-a*d^3/b)^(1/3)) + 2*(b*x^2*cosh(d*x + c 
)^2 - b*x^2*sinh(d*x + c)^2)*(a*d^3/b)^(1/3)*Ei(d*x + (a*d^3/b)^(1/3))*cos 
h(-c + (a*d^3/b)^(1/3)) - (a*d^3/b)^(1/3)*((sqrt(-3)*b*x^2 + b*x^2)*cosh(d 
*x + c)^2 - (sqrt(-3)*b*x^2 + b*x^2)*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)*b*x^2 + b*x^2)*cosh(d*x + c)^2 - (sqrt(-3)*b*x^2 
 + b*x^2)*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(...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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