3.1.41 \(\int \frac {\sinh (b x) \text {Shi}(b x)}{x} \, dx\) [41]

Optimal. Leaf size=10 \[ \frac {\text {Shi}(b x)^2}{2} \]

[Out]

1/2*Shi(b*x)^2

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6818} \begin {gather*} \frac {\text {Shi}(b x)^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sinh[b*x]*SinhIntegral[b*x])/x,x]

[Out]

SinhIntegral[b*x]^2/2

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {\text {Shi}(b x)^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sinh[b*x]*SinhIntegral[b*x])/x,x]

[Out]

SinhIntegral[b*x]^2/2

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Maple [A]
time = 0.07, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {\hyperbolicSineIntegral \left (b x \right )^{2}}{2}\)	\(9\)
default	\(\frac {\hyperbolicSineIntegral \left (b x \right )^{2}}{2}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*Shi(b*x)^2

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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,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [A]
time = 0.23, size = 7, normalized size = 0.70 \begin {gather*} \frac {\operatorname {Shi}^{2}{\left (b x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Shi(b*x)**2/2

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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,x, algorithm="giac")

[Out]

integrate(Shi(b*x)*sinh(b*x)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.10 \begin {gather*} \frac {{\mathrm {sinhint}\left (b\,x\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinhint(b*x)^2/2

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