3.1.42 \(\int \sinh (b x) \text {Shi}(b x) \, dx\) [42]

Optimal. Leaf size=25 \[ \frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \]

[Out]

cosh(b*x)*Shi(b*x)/b-1/2*Shi(2*b*x)/b

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Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {6675, 12, 5556, 3379} \begin {gather*} \frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[b*x]*SinhIntegral[b*x])/b - SinhIntegral[2*b*x]/(2*b)

Rule 12

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

Rule 3379

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

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 6675

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

Rubi steps

\begin {align*} \int \sinh (b x) \text {Shi}(b x) \, dx &=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx\\ &=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b}\\ &=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{2 x} \, dx}{b}\\ &=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{x} \, dx}{2 b}\\ &=\frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[b*x]*SinhIntegral[b*x])/b - SinhIntegral[2*b*x]/(2*b)

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Maple [A]
time = 0.16, size = 22, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\cosh \left (b x \right ) \hyperbolicSineIntegral \left (b x \right )-\frac {\hyperbolicSineIntegral \left (2 b x \right )}{2}}{b}\)	\(22\)
default	\(\frac {\cosh \left (b x \right ) \hyperbolicSineIntegral \left (b x \right )-\frac {\hyperbolicSineIntegral \left (2 b x \right )}{2}}{b}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Shi(b*x)*sinh(b*x),x,method=_RETURNVERBOSE)

[Out]

1/b*(cosh(b*x)*Shi(b*x)-1/2*Shi(2*b*x))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x)*sinh(b*x), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sinh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(b*x)*Shi(b*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x)*sinh(b*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \mathrm {sinhint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinhint(b*x)*sinh(b*x),x)

[Out]

int(sinhint(b*x)*sinh(b*x), x)

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