3.1.49 \(\int \cosh (b x) \text {Shi}(b x) \, dx\) [49]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 34, 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 = {6681, 12, 3393, 3382} \begin {gather*} -\frac {\text {Chi}(2 b x)}{2 b}+\frac {\text {Shi}(b x) \sinh (b x)}{b}+\frac {\log (x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*CoshIntegral[2*b*x]/b + Log[x]/(2*b) + (Sinh[b*x]*SinhIntegral[b*x])/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 3382

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]

Rule 3393

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 6681

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]

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*}

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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.06 \begin {gather*} -\frac {\text {Chi}(2 b x)}{2 b}+\frac {\log (b x)}{2 b}+\frac {\sinh (b x) \text {Shi}(b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*CoshIntegral[2*b*x]/b + Log[b*x]/(2*b) + (Sinh[b*x]*SinhIntegral[b*x])/b

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Maple [A]
time = 0.31, size = 28, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\hyperbolicSineIntegral \left (b x \right ) \sinh \left (b x \right )+\frac {\ln \left (b x \right )}{2}-\frac {\hyperbolicCosineIntegral \left (2 b x \right )}{2}}{b}\)	\(28\)
default	\(\frac {\hyperbolicSineIntegral \left (b x \right ) \sinh \left (b x \right )+\frac {\ln \left (b x \right )}{2}-\frac {\hyperbolicCosineIntegral \left (2 b x \right )}{2}}{b}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(Shi(b*x)*sinh(b*x)+1/2*ln(b*x)-1/2*Chi(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(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 = 0.00, 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cosh {\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(cosh(b*x)*Shi(b*x),x)

[Out]

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

[Out]

integrate(Shi(b*x)*cosh(b*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinhint(b*x)*cosh(b*x), x)

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