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 = 0.026271, antiderivative size = 35, normalized size of antiderivative = 1., 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 6532
Rule 12
Rule 3296
Rule 2637
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*}
Mathematica [A] time = 0.0073354, size = 35, normalized size = 1. \[ \frac{\sinh (b x)}{2 b^2}+\frac{1}{2} x^2 \text{Shi}(b x)-\frac{x \cosh (b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 32, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{b}^{2}{x}^{2}{\it Shi} \left ( bx \right ) }{2}}-{\frac{bx\cosh \left ( bx \right ) }{2}}+{\frac{\sinh \left ( bx \right ) }{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Shi}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \operatorname{Shi}\left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.774986, size = 29, normalized size = 0.83 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Shi}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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