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

Optimal. Leaf size=34 \[ -\frac{\text{Chi}(2 b x)}{2 b}+\frac{\text{Shi}(b x) \sinh (b x)}{b}+\frac{\log (x)}{2 b} \]

[Out]

-CoshIntegral[2*b*x]/(2*b) + Log[x]/(2*b) + (Sinh[b*x]*SinhIntegral[b*x])/b

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Rubi [A]  time = 0.0602898, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6546, 12, 3312, 3301} \[ -\frac{\text{Chi}(2 b x)}{2 b}+\frac{\text{Shi}(b x) \sinh (b x)}{b}+\frac{\log (x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[b*x]*SinhIntegral[b*x],x]

[Out]

-CoshIntegral[2*b*x]/(2*b) + Log[x]/(2*b) + (Sinh[b*x]*SinhIntegral[b*x])/b

Rule 6546

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \cosh (b x) \text{Shi}(b x) \, dx &=\frac{\sinh (b x) \text{Shi}(b x)}{b}-\int \frac{\sinh ^2(b x)}{b x} \, dx\\ &=\frac{\sinh (b x) \text{Shi}(b x)}{b}-\frac{\int \frac{\sinh ^2(b x)}{x} \, dx}{b}\\ &=\frac{\sinh (b x) \text{Shi}(b x)}{b}+\frac{\int \left (\frac{1}{2 x}-\frac{\cosh (2 b x)}{2 x}\right ) \, dx}{b}\\ &=\frac{\log (x)}{2 b}+\frac{\sinh (b x) \text{Shi}(b x)}{b}-\frac{\int \frac{\cosh (2 b x)}{x} \, dx}{2 b}\\ &=-\frac{\text{Chi}(2 b x)}{2 b}+\frac{\log (x)}{2 b}+\frac{\sinh (b x) \text{Shi}(b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0126236, size = 36, normalized size = 1.06 \[ -\frac{\text{Chi}(2 b x)}{2 b}+\frac{\text{Shi}(b x) \sinh (b x)}{b}+\frac{\log (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[b*x]*SinhIntegral[b*x],x]

[Out]

-CoshIntegral[2*b*x]/(2*b) + Log[b*x]/(2*b) + (Sinh[b*x]*SinhIntegral[b*x])/b

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Maple [A]  time = 0.052, size = 33, normalized size = 1. \begin{align*}{\frac{{\it Shi} \left ( bx \right ) \sinh \left ( bx \right ) }{b}}+{\frac{\ln \left ( bx \right ) }{2\,b}}-{\frac{{\it Chi} \left ( 2\,bx \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x)*Shi(b*x),x)

[Out]

Shi(b*x)*sinh(b*x)/b+1/2/b*ln(b*x)-1/2*Chi(2*b*x)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x)*cosh(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh(b*x)*sinh_integral(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x)*Shi(b*x),x)

[Out]

Integral(cosh(b*x)*Shi(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x)*cosh(b*x), x)