Optimal. Leaf size=49 \[ \frac{2 x \sinh (b x)}{3 b^2}-\frac{2 \cosh (b x)}{3 b^3}+\frac{1}{3} x^3 \text{Shi}(b x)-\frac{x^2 \cosh (b x)}{3 b} \]
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Rubi [A] time = 0.0559646, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6532, 12, 3296, 2638} \[ \frac{2 x \sinh (b x)}{3 b^2}-\frac{2 \cosh (b x)}{3 b^3}+\frac{1}{3} x^3 \text{Shi}(b x)-\frac{x^2 \cosh (b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 6532
Rule 12
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^2 \text{Shi}(b x) \, dx &=\frac{1}{3} x^3 \text{Shi}(b x)-\frac{1}{3} b \int \frac{x^2 \sinh (b x)}{b} \, dx\\ &=\frac{1}{3} x^3 \text{Shi}(b x)-\frac{1}{3} \int x^2 \sinh (b x) \, dx\\ &=-\frac{x^2 \cosh (b x)}{3 b}+\frac{1}{3} x^3 \text{Shi}(b x)+\frac{2 \int x \cosh (b x) \, dx}{3 b}\\ &=-\frac{x^2 \cosh (b x)}{3 b}+\frac{2 x \sinh (b x)}{3 b^2}+\frac{1}{3} x^3 \text{Shi}(b x)-\frac{2 \int \sinh (b x) \, dx}{3 b^2}\\ &=-\frac{2 \cosh (b x)}{3 b^3}-\frac{x^2 \cosh (b x)}{3 b}+\frac{2 x \sinh (b x)}{3 b^2}+\frac{1}{3} x^3 \text{Shi}(b x)\\ \end{align*}
Mathematica [A] time = 0.0253652, size = 44, normalized size = 0.9 \[ -\frac{\left (b^2 x^2+2\right ) \cosh (b x)}{3 b^3}+\frac{2 x \sinh (b x)}{3 b^2}+\frac{1}{3} x^3 \text{Shi}(b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 44, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3}{\it Shi} \left ( bx \right ) }{3}}-{\frac{{b}^{2}{x}^{2}\cosh \left ( bx \right ) }{3}}+{\frac{2\,bx\sinh \left ( bx \right ) }{3}}-{\frac{2\,\cosh \left ( bx \right ) }{3}} \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^{2}{\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^{2} \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 = 1.47954, size = 46, normalized size = 0.94 \begin{align*} \frac{x^{3} \operatorname{Shi}{\left (b x \right )}}{3} - \frac{x^{2} \cosh{\left (b x \right )}}{3 b} + \frac{2 x \sinh{\left (b x \right )}}{3 b^{2}} - \frac{2 \cosh{\left (b x \right )}}{3 b^{3}} \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^{2}{\rm Shi}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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