3.5 \(\int \text{Shi}(b x) \, dx\)

Optimal. Leaf size=16 \[ x \text{Shi}(b x)-\frac{\cosh (b x)}{b} \]

[Out]

-(Cosh[b*x]/b) + x*SinhIntegral[b*x]

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Rubi [A]  time = 0.0041108, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6528} \[ x \text{Shi}(b x)-\frac{\cosh (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[SinhIntegral[b*x],x]

[Out]

-(Cosh[b*x]/b) + x*SinhIntegral[b*x]

Rule 6528

Int[SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*SinhIntegral[a + b*x])/b, x] - Simp[Cosh[a
+ b*x]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \text{Shi}(b x) \, dx &=-\frac{\cosh (b x)}{b}+x \text{Shi}(b x)\\ \end{align*}

Mathematica [A]  time = 0.001409, size = 16, normalized size = 1. \[ x \text{Shi}(b x)-\frac{\cosh (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[SinhIntegral[b*x],x]

[Out]

-(Cosh[b*x]/b) + x*SinhIntegral[b*x]

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Maple [A]  time = 0.049, size = 19, normalized size = 1.2 \begin{align*}{\frac{bx{\it Shi} \left ( bx \right ) -\cosh \left ( bx \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Shi(b*x),x)

[Out]

1/b*(b*x*Shi(b*x)-cosh(b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x),x, algorithm="maxima")

[Out]

integrate(Shi(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x),x, algorithm="fricas")

[Out]

integral(sinh_integral(b*x), x)

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Sympy [A]  time = 0.842919, size = 12, normalized size = 0.75 \begin{align*} x \operatorname{Shi}{\left (b x \right )} - \frac{\cosh{\left (b x \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x),x)

[Out]

x*Shi(b*x) - cosh(b*x)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x),x, algorithm="giac")

[Out]

integrate(Shi(b*x), x)