\(\int \frac {\cosh (c+d x)}{x (a+b x^3)} \, dx\) [99]

Optimal result

Integrand size = 19, antiderivative size = 303 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )} \, dx=\frac {\cosh (c) \text {Chi}(d x)}{a}-\frac {\cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a}+\frac {\sinh (c) \text {Shi}(d x)}{a}+\frac {\sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a}-\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.13 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.61 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )} \, dx=-\frac {-6 \cosh (c) \text {Chi}(d x)+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))-\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})-\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+\text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+\cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+\sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))\&\right ]-6 \sinh (c) \text {Shi}(d x)}{6 a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 303, 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.

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

\(\Big \downarrow \) 5816

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Cosh[c]*CoshIntegral[d*x])/a - (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) - (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) - (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b 
^(1/3) + d*x])/(3*a) + (Sinh[c]*SinhIntegral[d*x])/a + (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) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + 
 d*x])/(3*a) - (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)
 

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 = 0.55 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.46

method	result	size
risch	\(-\frac {{\mathrm e}^{-c} \operatorname {expIntegral}_{1}\left (d x \right )}{2 a}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }{\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (d x -\textit {\_R1} +c \right )}{6 a}-\frac {{\mathrm e}^{c} \operatorname {expIntegral}_{1}\left (-d x \right )}{2 a}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +d^{3} a -b \,c^{3}\right )}{\sum }{\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-d x +\textit {\_R1} -c \right )}{6 a}\)	\(138\)

Input:

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

Output:

-1/2/a*exp(-c)*Ei(1,d*x)+1/6/a*sum(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/a*exp(c)*Ei(1,-d*x)+1/6/a*sum 
(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. 530 vs. \(2 (227) = 454\).

Time = 0.12 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.75 \[ \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(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) + Ei(-d*x - 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cos 
h(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1) - c) + 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) + 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) + Ei(-d*x + (-a*d^3/b)^(1/3))*cosh(c + (-a*d^3/b)^(1/3)) - 3*(Ei(d 
*x) + Ei(-d*x))*cosh(c) + Ei(d*x + (a*d^3/b)^(1/3))*cosh(-c + (a*d^3/b)^(1 
/3)) + 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) + Ei(-d*x - 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*si 
nh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1) - c) - 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) - 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) - Ei(-d*x + (-a*d^3/b)^(1/3))*sinh(c + (-a*d^3/b)^(1/3)) - 3*(Ei( 
d*x) - Ei(-d*x))*sinh(c) - Ei(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^( 
1/3)))/a
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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