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

Optimal. Leaf size=48 \[ b \text{CannotIntegrate}\left (\frac{\text{Shi}(b x) \cosh (b x)}{x},x\right )+b \text{Shi}(2 b x)-\frac{\text{Shi}(b x) \sinh (b x)}{x}-\frac{\sinh ^2(b x)}{x} \]

[Out]

b*CannotIntegrate[(Cosh[b*x]*SinhIntegral[b*x])/x, x] - Sinh[b*x]^2/x - (Sinh[b*x]*SinhIntegral[b*x])/x + b*Si
nhIntegral[2*b*x]

________________________________________________________________________________________

Rubi [A]  time = 0.15102, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sinh (b x) \text{Shi}(b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\sinh (b x) \text{Shi}(b x)}{x^2} \, dx &=-\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 [A]  time = 0.25221, size = 0, normalized size = 0. \[ \int \frac{\sinh (b x) \text{Shi}(b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Shi} \left ( bx \right ) \sinh \left ( bx \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x)*sinh(b*x)/x^2, x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sinh \left (b x\right ) \operatorname{Shi}\left (b x\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (b x \right )} \operatorname{Shi}{\left (b x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(b*x)*Shi(b*x)/x**2, x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left (b x\right ) \sinh \left (b x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x)*sinh(b*x)/x^2, x)