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

Optimal. Leaf size=46 \[ -\frac{\text{Chi}(2 a+2 b x)}{2 b}+\frac{\text{Shi}(a+b x) \sinh (a+b x)}{b}+\frac{\log (a+b x)}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.075279, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {6546, 3312, 3301} \[ -\frac{\text{Chi}(2 a+2 b x)}{2 b}+\frac{\text{Shi}(a+b x) \sinh (a+b x)}{b}+\frac{\log (a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-CoshIntegral[2*a + 2*b*x]/(2*b) + Log[a + b*x]/(2*b) + (Sinh[a + b*x]*SinhIntegral[a + 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 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 (a+b x) \text{Shi}(a+b x) \, dx &=\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b}-\int \frac{\sinh ^2(a+b x)}{a+b x} \, dx\\ &=\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b}+\int \left (\frac{1}{2 (a+b x)}-\frac{\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx\\ &=\frac{\log (a+b x)}{2 b}+\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b}-\frac{1}{2} \int \frac{\cosh (2 a+2 b x)}{a+b x} \, dx\\ &=-\frac{\text{Chi}(2 a+2 b x)}{2 b}+\frac{\log (a+b x)}{2 b}+\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0247566, size = 45, normalized size = 0.98 \[ -\frac{\text{Chi}(2 (a+b x))}{2 b}+\frac{\text{Shi}(a+b x) \sinh (a+b x)}{b}+\frac{\log (a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.051, size = 43, normalized size = 0.9 \begin{align*} -{\frac{{\it Chi} \left ( 2\,bx+2\,a \right ) }{2\,b}}+{\frac{\ln \left ( bx+a \right ) }{2\,b}}+{\frac{{\it Shi} \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(a + b*x)*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 ) \cosh \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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