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(b*x+a)/b+(b*x+a)*Shi(b*x+a)/b

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, 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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fricas [F]  time = 2.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Shi}\left (b x + a\right ), x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Shi}\left (b x + a\right )\,{d x} \]

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)

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maple [A]  time = 0.01, size = 26, normalized size = 0.96 \[ \frac {\left (b x +a \right ) \Shi \left (b x +a \right )-\cosh \left (b x +a \right )}{b} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Shi}\left (b x + a\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ x\,\mathrm {sinhint}\left (a+b\,x\right )-\frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}+\frac {a\,\mathrm {sinhint}\left (a+b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinhint(a + b*x),x)

[Out]

x*sinhint(a + b*x) - exp(a + b*x)/(2*b) - exp(- a - b*x)/(2*b) + (a*sinhint(a + b*x))/b

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {Shi}{\left (a + b x \right )}\, dx \]

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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