Optimal. Leaf size=35 \[ \frac {\sinh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(b x)-\frac {x \cosh (b x)}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6532, 12, 3296, 2637} \[ \frac {\sinh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(b x)-\frac {x \cosh (b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2637
Rule 3296
Rule 6532
Rubi steps
\begin {align*} \int x \text {Shi}(b x) \, dx &=\frac {1}{2} x^2 \text {Shi}(b x)-\frac {1}{2} b \int \frac {x \sinh (b x)}{b} \, dx\\ &=\frac {1}{2} x^2 \text {Shi}(b x)-\frac {1}{2} \int x \sinh (b x) \, dx\\ &=-\frac {x \cosh (b x)}{2 b}+\frac {1}{2} x^2 \text {Shi}(b x)+\frac {\int \cosh (b x) \, dx}{2 b}\\ &=-\frac {x \cosh (b x)}{2 b}+\frac {\sinh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(b x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ \frac {\sinh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(b x)-\frac {x \cosh (b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Shi}\left (b x\right ), x\right ) \]
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 \[ \int x {\rm Shi}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 32, normalized size = 0.91 \[ \frac {\frac {b^{2} x^{2} \Shi \left (b x \right )}{2}-\frac {b x \cosh \left (b x \right )}{2}+\frac {\sinh \left (b x \right )}{2}}{b^{2}} \]
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 \[ \int x {\rm Shi}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \frac {\frac {\mathrm {sinh}\left (b\,x\right )}{2}-\frac {b\,x\,\mathrm {cosh}\left (b\,x\right )}{2}}{b^2}+\frac {x^2\,\mathrm {sinhint}\left (b\,x\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 29, normalized size = 0.83 \[ \frac {x^{2} \operatorname {Shi}{\left (b x \right )}}{2} - \frac {x \cosh {\left (b x \right )}}{2 b} + \frac {\sinh {\left (b x \right )}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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