Optimal. Leaf size=63 \[ \frac{3 x^2 \sinh (b x)}{4 b^2}+\frac{3 \sinh (b x)}{2 b^4}-\frac{3 x \cosh (b x)}{2 b^3}+\frac{1}{4} x^4 \text{Shi}(b x)-\frac{x^3 \cosh (b x)}{4 b} \]
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Rubi [A] time = 0.0839202, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6532, 12, 3296, 2637} \[ \frac{3 x^2 \sinh (b x)}{4 b^2}+\frac{3 \sinh (b x)}{2 b^4}-\frac{3 x \cosh (b x)}{2 b^3}+\frac{1}{4} x^4 \text{Shi}(b x)-\frac{x^3 \cosh (b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 6532
Rule 12
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 \text{Shi}(b x) \, dx &=\frac{1}{4} x^4 \text{Shi}(b x)-\frac{1}{4} b \int \frac{x^3 \sinh (b x)}{b} \, dx\\ &=\frac{1}{4} x^4 \text{Shi}(b x)-\frac{1}{4} \int x^3 \sinh (b x) \, dx\\ &=-\frac{x^3 \cosh (b x)}{4 b}+\frac{1}{4} x^4 \text{Shi}(b x)+\frac{3 \int x^2 \cosh (b x) \, dx}{4 b}\\ &=-\frac{x^3 \cosh (b x)}{4 b}+\frac{3 x^2 \sinh (b x)}{4 b^2}+\frac{1}{4} x^4 \text{Shi}(b x)-\frac{3 \int x \sinh (b x) \, dx}{2 b^2}\\ &=-\frac{3 x \cosh (b x)}{2 b^3}-\frac{x^3 \cosh (b x)}{4 b}+\frac{3 x^2 \sinh (b x)}{4 b^2}+\frac{1}{4} x^4 \text{Shi}(b x)+\frac{3 \int \cosh (b x) \, dx}{2 b^3}\\ &=-\frac{3 x \cosh (b x)}{2 b^3}-\frac{x^3 \cosh (b x)}{4 b}+\frac{3 \sinh (b x)}{2 b^4}+\frac{3 x^2 \sinh (b x)}{4 b^2}+\frac{1}{4} x^4 \text{Shi}(b x)\\ \end{align*}
Mathematica [A] time = 0.0361031, size = 53, normalized size = 0.84 \[ \frac{3 \left (b^2 x^2+2\right ) \sinh (b x)}{4 b^4}-\frac{x \left (b^2 x^2+6\right ) \cosh (b x)}{4 b^3}+\frac{1}{4} x^4 \text{Shi}(b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 56, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{{b}^{4}{x}^{4}{\it Shi} \left ( bx \right ) }{4}}-{\frac{{b}^{3}{x}^{3}\cosh \left ( bx \right ) }{4}}+{\frac{3\,{b}^{2}{x}^{2}\sinh \left ( bx \right ) }{4}}-{\frac{3\,bx\cosh \left ( bx \right ) }{2}}+{\frac{3\,\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^{3}{\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^{3} \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 = 1.26648, size = 61, normalized size = 0.97 \begin{align*} \frac{x^{4} \operatorname{Shi}{\left (b x \right )}}{4} - \frac{x^{3} \cosh{\left (b x \right )}}{4 b} + \frac{3 x^{2} \sinh{\left (b x \right )}}{4 b^{2}} - \frac{3 x \cosh{\left (b x \right )}}{2 b^{3}} + \frac{3 \sinh{\left (b x \right )}}{2 b^{4}} \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^{3}{\rm Shi}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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