Optimal. Leaf size=25 \[ \frac {\cosh (b x) \text {Shi}(b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b} \]
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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.
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Rule 12
Rule 3379
Rule 5556
Rule 6675
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.
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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.
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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.
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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.
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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.
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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.
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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.
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