\(\int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx\) [40]

Optimal result

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

[Out]

b*CannotIntegrate(cosh(b*x)*Shi(b*x)/x,x)+b*Shi(2*b*x)-Shi(b*x)*sinh(b*x)/x-sinh(b*x)^2/x

Rubi [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx=\int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx \]

[In]

Int[(Sinh[b*x]*SinhIntegral[b*x])/x^2,x]

[Out]

-(Sinh[b*x]^2/x) - (Sinh[b*x]*SinhIntegral[b*x])/x + b*SinhIntegral[2*b*x] + b*Defer[Int][(Cosh[b*x]*SinhInteg
ral[b*x])/x, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sinh (b x) \text {Shi}(b x)}{x}+b \int \frac {\sinh ^2(b x)}{b x^2} \, dx+b \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ & = -\frac {\sinh (b x) \text {Shi}(b x)}{x}+b \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx+\int \frac {\sinh ^2(b x)}{x^2} \, dx \\ & = -\frac {\sinh ^2(b x)}{x}-\frac {\sinh (b x) \text {Shi}(b x)}{x}-(2 i b) \int \frac {i \sinh (2 b x)}{2 x} \, dx+b \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ & = -\frac {\sinh ^2(b x)}{x}-\frac {\sinh (b x) \text {Shi}(b x)}{x}+b \int \frac {\sinh (2 b x)}{x} \, dx+b \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ & = -\frac {\sinh ^2(b x)}{x}-\frac {\sinh (b x) \text {Shi}(b x)}{x}+b \text {Shi}(2 b x)+b \int \frac {\cosh (b x) \text {Shi}(b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx=\int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx \]

[In]

Integrate[(Sinh[b*x]*SinhIntegral[b*x])/x^2,x]

[Out]

Integrate[(Sinh[b*x]*SinhIntegral[b*x])/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {Shi}\left (b x \right ) \sinh \left (b x \right )}{x^{2}}d x\]

[In]

int(Shi(b*x)*sinh(b*x)/x^2,x)

[Out]

int(Shi(b*x)*sinh(b*x)/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \]

[In]

integrate(Shi(b*x)*sinh(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(sinh(b*x)*sinh_integral(b*x)/x^2, x)

Sympy [N/A]

Not integrable

Time = 2.38 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx=\int \frac {\sinh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}}{x^{2}}\, dx \]

[In]

integrate(Shi(b*x)*sinh(b*x)/x**2,x)

[Out]

Integral(sinh(b*x)*Shi(b*x)/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \]

[In]

integrate(Shi(b*x)*sinh(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(Shi(b*x)*sinh(b*x)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}} \,d x } \]

[In]

integrate(Shi(b*x)*sinh(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(Shi(b*x)*sinh(b*x)/x^2, x)

Mupad [N/A]

Not integrable

Time = 4.85 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\sinh (b x) \text {Shi}(b x)}{x^2} \, dx=\int \frac {\mathrm {sinhint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right )}{x^2} \,d x \]

[In]

int((sinhint(b*x)*sinh(b*x))/x^2,x)

[Out]

int((sinhint(b*x)*sinh(b*x))/x^2, x)