3.21 \(\int \text{Shi}(a+b x) \, dx\)

Optimal. Leaf size=27 \[ \frac{(a+b x) \text{Shi}(a+b x)}{b}-\frac{\cosh (a+b x)}{b} \]

[Out]

-(Cosh[a + b*x]/b) + ((a + b*x)*SinhIntegral[a + b*x])/b

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Rubi [A]  time = 0.0055133, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6528} \[ \frac{(a+b x) \text{Shi}(a+b x)}{b}-\frac{\cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[SinhIntegral[a + b*x],x]

[Out]

-(Cosh[a + b*x]/b) + ((a + b*x)*SinhIntegral[a + b*x])/b

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

Mathematica [A]  time = 0.0348254, size = 42, normalized size = 1.56 \[ x \text{Shi}(a+b x)+\frac{a \text{Shi}(a+b x)}{b}-\frac{\sinh (a) \sinh (b x)}{b}-\frac{\cosh (a) \cosh (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[SinhIntegral[a + b*x],x]

[Out]

-((Cosh[a]*Cosh[b*x])/b) - (Sinh[a]*Sinh[b*x])/b + (a*SinhIntegral[a + b*x])/b + x*SinhIntegral[a + b*x]

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Maple [A]  time = 0.044, size = 26, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ){\it Shi} \left ( bx+a \right ) -\cosh \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Shi(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sinh_integral(b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Shi}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x+a),x)

[Out]

Integral(Shi(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x + a), x)