\(\int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx\) [46]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=-\frac {b \cosh (2 b x)}{4 x}-\frac {b \sinh ^2(b x)}{2 x}-\frac {\sinh (2 b x)}{8 x^2}-\frac {\cosh (b x) \text {Shi}(b x)}{2 x^2}-\frac {b \sinh (b x) \text {Shi}(b x)}{2 x}+b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \text {Int}\left (\frac {\cosh (b x) \text {Shi}(b x)}{x},x\right ) \] Output:

-1/4*b*cosh(2*b*x)/x-1/2*b*sinh(b*x)^2/x-1/8*sinh(2*b*x)/x^2-1/2*cosh(b*x) 
*Shi(b*x)/x^2-1/2*b*sinh(b*x)*Shi(b*x)/x+b^2*Shi(2*b*x)+1/2*b^2*Defer(Int) 
(cosh(b*x)*Shi(b*x)/x,x)
 

Mathematica [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx \] Input:

Integrate[(Cosh[b*x]*SinhIntegral[b*x])/x^3,x]
 

Output:

Integrate[(Cosh[b*x]*SinhIntegral[b*x])/x^3, x]
 

Rubi [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

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.

\(\displaystyle \int \frac {\text {Shi}(b x) \cosh (b x)}{x^3} \, dx\)

\(\Big \downarrow \) 7104

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{2} b \int \frac {\cosh (b x) \sinh (b x)}{b x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{2} \int \frac {\cosh (b x) \sinh (b x)}{x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{2} \int \frac {\sinh (2 b x)}{2 x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{4} \int \frac {\sinh (2 b x)}{x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx+\frac {1}{4} \int -\frac {i \sin (2 i b x)}{x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \int \frac {\sin (2 i b x)}{x^3}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \int \frac {\cosh (2 b x)}{x^2}dx-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \int \frac {\sin \left (2 i b x+\frac {\pi }{2}\right )}{x^2}dx-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \left (-\frac {\cosh (2 b x)}{x}+2 i b \int -\frac {i \sinh (2 b x)}{x}dx\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \left (2 b \int \frac {\sinh (2 b x)}{x}dx-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \left (-\frac {\cosh (2 b x)}{x}+2 b \int -\frac {i \sin (2 i b x)}{x}dx\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {1}{4} i \left (i b \left (-\frac {\cosh (2 b x)}{x}-2 i b \int \frac {\sin (2 i b x)}{x}dx\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {1}{2} b \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2}dx-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 7098

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \int \frac {\sinh ^2(b x)}{b x^2}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+\int \frac {\sinh ^2(b x)}{x^2}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+\int -\frac {\sin (i b x)^2}{x^2}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx-\int \frac {\sin (i b x)^2}{x^2}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3794

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx-2 i b \int \frac {i \sinh (2 b x)}{2 x}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \int \frac {\sinh (2 b x)}{x}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \int -\frac {i \sin (2 i b x)}{x}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx-i b \int \frac {\sin (2 i b x)}{x}dx-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \text {Shi}(2 b x)-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {1}{2} b \left (b \int \frac {\cosh (b x) \text {Shi}(b x)}{x}dx+b \text {Shi}(2 b x)-\frac {\text {Shi}(b x) \sinh (b x)}{x}-\frac {\sinh ^2(b x)}{x}\right )-\frac {\text {Shi}(b x) \cosh (b x)}{2 x^2}-\frac {1}{4} i \left (i b \left (2 b \text {Shi}(2 b x)-\frac {\cosh (2 b x)}{x}\right )-\frac {i \sinh (2 b x)}{2 x^2}\right )\)

Input:

Int[(Cosh[b*x]*SinhIntegral[b*x])/x^3,x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \frac {\cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )}{x^{3}}d x\]

Input:

int(cosh(b*x)*Shi(b*x)/x^3,x)
 

Output:

int(cosh(b*x)*Shi(b*x)/x^3,x)
 

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="fricas")
 

Output:

integral(cosh(b*x)*sinh_integral(b*x)/x^3, x)
 

Sympy [N/A]

Not integrable

Time = 3.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{3}}\, dx \] Input:

integrate(cosh(b*x)*Shi(b*x)/x**3,x)
 

Output:

Integral(cosh(b*x)*Shi(b*x)/x**3, x)
 

Maxima [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="maxima")
 

Output:

integrate(Shi(b*x)*cosh(b*x)/x^3, x)
 

Giac [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(cosh(b*x)*Shi(b*x)/x^3,x, algorithm="giac")
 

Output:

integrate(Shi(b*x)*cosh(b*x)/x^3, x)
 

Mupad [N/A]

Not integrable

Time = 4.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^3} \,d x \] Input:

int((sinhint(b*x)*cosh(b*x))/x^3,x)
 

Output:

int((sinhint(b*x)*cosh(b*x))/x^3, x)
 

Reduce [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Shi}(b x)}{x^3} \, dx=\int \frac {\cosh \left (b x \right ) \mathit {shi} \left (b x \right )}{x^{3}}d x \] Input:

int(cosh(b*x)*Shi(b*x)/x^3,x)
 

Output:

int((cosh(b*x)*shi(b*x))/x**3,x)