Optimal. Leaf size=128 \[ -\frac{3 x^2 \text{Shi}(b x) \cosh (b x)}{b^2}+\frac{3 \text{Shi}(2 b x)}{b^4}+\frac{6 x \text{Shi}(b x) \sinh (b x)}{b^3}-\frac{6 \text{Shi}(b x) \cosh (b x)}{b^4}-\frac{x^2 \sinh (b x) \cosh (b x)}{2 b^2}+\frac{4 x}{b^3}+\frac{2 x \sinh ^2(b x)}{b^3}-\frac{4 \sinh (b x) \cosh (b x)}{b^4}+\frac{x^3 \text{Shi}(b x) \sinh (b x)}{b}+\frac{x^3}{6 b} \]
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Rubi [A] time = 0.189259, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6548, 12, 3311, 30, 2635, 8, 6542, 5372, 6540, 5448, 3298} \[ -\frac{3 x^2 \text{Shi}(b x) \cosh (b x)}{b^2}+\frac{3 \text{Shi}(2 b x)}{b^4}+\frac{6 x \text{Shi}(b x) \sinh (b x)}{b^3}-\frac{6 \text{Shi}(b x) \cosh (b x)}{b^4}-\frac{x^2 \sinh (b x) \cosh (b x)}{2 b^2}+\frac{4 x}{b^3}+\frac{2 x \sinh ^2(b x)}{b^3}-\frac{4 \sinh (b x) \cosh (b x)}{b^4}+\frac{x^3 \text{Shi}(b x) \sinh (b x)}{b}+\frac{x^3}{6 b} \]
Antiderivative was successfully verified.
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Rule 6548
Rule 12
Rule 3311
Rule 30
Rule 2635
Rule 8
Rule 6542
Rule 5372
Rule 6540
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int x^3 \cosh (b x) \text{Shi}(b x) \, dx &=\frac{x^3 \sinh (b x) \text{Shi}(b x)}{b}-\frac{3 \int x^2 \sinh (b x) \text{Shi}(b x) \, dx}{b}-\int \frac{x^2 \sinh ^2(b x)}{b} \, dx\\ &=-\frac{3 x^2 \cosh (b x) \text{Shi}(b x)}{b^2}+\frac{x^3 \sinh (b x) \text{Shi}(b x)}{b}+\frac{6 \int x \cosh (b x) \text{Shi}(b x) \, dx}{b^2}-\frac{\int x^2 \sinh ^2(b x) \, dx}{b}+\frac{3 \int \frac{x \cosh (b x) \sinh (b x)}{b} \, dx}{b}\\ &=-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac{x \sinh ^2(b x)}{2 b^3}-\frac{3 x^2 \cosh (b x) \text{Shi}(b x)}{b^2}+\frac{6 x \sinh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \sinh (b x) \text{Shi}(b x)}{b}-\frac{\int \sinh ^2(b x) \, dx}{2 b^3}-\frac{6 \int \sinh (b x) \text{Shi}(b x) \, dx}{b^3}+\frac{3 \int x \cosh (b x) \sinh (b x) \, dx}{b^2}-\frac{6 \int \frac{\sinh ^2(b x)}{b} \, dx}{b^2}+\frac{\int x^2 \, dx}{2 b}\\ &=\frac{x^3}{6 b}-\frac{\cosh (b x) \sinh (b x)}{4 b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac{2 x \sinh ^2(b x)}{b^3}-\frac{6 \cosh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Shi}(b x)}{b^2}+\frac{6 x \sinh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \sinh (b x) \text{Shi}(b x)}{b}+\frac{\int 1 \, dx}{4 b^3}-\frac{3 \int \sinh ^2(b x) \, dx}{2 b^3}+\frac{6 \int \frac{\cosh (b x) \sinh (b x)}{b x} \, dx}{b^3}-\frac{6 \int \sinh ^2(b x) \, dx}{b^3}\\ &=\frac{x}{4 b^3}+\frac{x^3}{6 b}-\frac{4 \cosh (b x) \sinh (b x)}{b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac{2 x \sinh ^2(b x)}{b^3}-\frac{6 \cosh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Shi}(b x)}{b^2}+\frac{6 x \sinh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \sinh (b x) \text{Shi}(b x)}{b}+\frac{6 \int \frac{\cosh (b x) \sinh (b x)}{x} \, dx}{b^4}+\frac{3 \int 1 \, dx}{4 b^3}+\frac{3 \int 1 \, dx}{b^3}\\ &=\frac{4 x}{b^3}+\frac{x^3}{6 b}-\frac{4 \cosh (b x) \sinh (b x)}{b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac{2 x \sinh ^2(b x)}{b^3}-\frac{6 \cosh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Shi}(b x)}{b^2}+\frac{6 x \sinh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \sinh (b x) \text{Shi}(b x)}{b}+\frac{6 \int \frac{\sinh (2 b x)}{2 x} \, dx}{b^4}\\ &=\frac{4 x}{b^3}+\frac{x^3}{6 b}-\frac{4 \cosh (b x) \sinh (b x)}{b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac{2 x \sinh ^2(b x)}{b^3}-\frac{6 \cosh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Shi}(b x)}{b^2}+\frac{6 x \sinh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \sinh (b x) \text{Shi}(b x)}{b}+\frac{3 \int \frac{\sinh (2 b x)}{x} \, dx}{b^4}\\ &=\frac{4 x}{b^3}+\frac{x^3}{6 b}-\frac{4 \cosh (b x) \sinh (b x)}{b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac{2 x \sinh ^2(b x)}{b^3}-\frac{6 \cosh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Shi}(b x)}{b^2}+\frac{6 x \sinh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \sinh (b x) \text{Shi}(b x)}{b}+\frac{3 \text{Shi}(2 b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.102478, size = 94, normalized size = 0.73 \[ \frac{12 \text{Shi}(b x) \left (b x \left (b^2 x^2+6\right ) \sinh (b x)-3 \left (b^2 x^2+2\right ) \cosh (b x)\right )+2 b^3 x^3-3 b^2 x^2 \sinh (2 b x)+36 \text{Shi}(2 b x)+36 b x-24 \sinh (2 b x)+12 b x \cosh (2 b x)}{12 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 104, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{4}} \left ({\it Shi} \left ( bx \right ) \left ( \sinh \left ( bx \right ){b}^{3}{x}^{3}-3\,{b}^{2}{x}^{2}\cosh \left ( bx \right ) +6\,bx\sinh \left ( bx \right ) -6\,\cosh \left ( bx \right ) \right ) -{\frac{{b}^{2}{x}^{2}\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{2}}+{\frac{{x}^{3}{b}^{3}}{6}}+2\,bx \left ( \cosh \left ( bx \right ) \right ) ^{2}-4\,\cosh \left ( bx \right ) \sinh \left ( bx \right ) +2\,bx+3\,{\it Shi} \left ( 2\,bx \right ) \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 ) \cosh \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} \cosh \left (b x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \cosh{\left (b x \right )} \operatorname{Shi}{\left (b x \right )}\, dx \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 ) \cosh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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