\(\int \frac {x \cosh (c+d x)}{a+b x^3} \, dx\) [97]

Optimal result

Integrand size = 17, antiderivative size = 345 \[ \int \frac {x \cosh (c+d x)}{a+b x^3} \, dx=-\frac {(-1)^{2/3} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {(-1)^{2/3} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 \sqrt [3]{a} b^{2/3}} \] Output:

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

Mathematica [C] (verified)

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

Time = 5.05 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.52 \[ \int \frac {x \cosh (c+d x)}{a+b x^3} \, dx=\frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}}\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}}\&\right ]}{6 b} \] Input:

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

Output:

(RootSum[a + b*#1^3 & , (Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] - CoshInt 
egral[d*(x - #1)]*Sinh[c + d*#1] - Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] 
 + Sinh[c + d*#1]*SinhIntegral[d*(x - #1)])/#1 & ] + 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]*Sinh 
Integral[d*(x - #1)])/#1 & ])/(6*b)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/3*((-1)^(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])/(a^(1/3)*b^(2/3)) + ((-1)^(1/3)*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^(1/3)*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^(1/3)*b^(2/3)) + ((-1)^(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^(1/3)*b^(2/3)) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIn 
tegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*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^(1/3)*b^(2/3))
 

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 = 0.54 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.81

method	result	size
risch	\(\frac {d c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 b}+\frac {d c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 b}-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\textit {\_R1} \,{\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 b}-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }\frac {\textit {\_R1} \,{\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 b}\)	\(280\)

Input:

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

Output:

1/6*d/b*c*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*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*sum(_R1/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R 
1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/6*d/b*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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (237) = 474\).

Time = 0.12 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.94 \[ \int \frac {x \cosh (c+d x)}{a+b x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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