Optimal. Leaf size=90 \[ -\frac{\text{Shi}(2 b x)}{b^3}-\frac{2 x \text{Shi}(b x) \sinh (b x)}{b^2}+\frac{2 \text{Shi}(b x) \cosh (b x)}{b^3}-\frac{5 x}{4 b^2}-\frac{x \sinh ^2(b x)}{2 b^2}+\frac{5 \sinh (b x) \cosh (b x)}{4 b^3}+\frac{x^2 \text{Shi}(b x) \cosh (b x)}{b} \]
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Rubi [A] time = 0.128153, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6542, 12, 5372, 2635, 8, 6548, 6540, 5448, 3298} \[ -\frac{\text{Shi}(2 b x)}{b^3}-\frac{2 x \text{Shi}(b x) \sinh (b x)}{b^2}+\frac{2 \text{Shi}(b x) \cosh (b x)}{b^3}-\frac{5 x}{4 b^2}-\frac{x \sinh ^2(b x)}{2 b^2}+\frac{5 \sinh (b x) \cosh (b x)}{4 b^3}+\frac{x^2 \text{Shi}(b x) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6542
Rule 12
Rule 5372
Rule 2635
Rule 8
Rule 6548
Rule 6540
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int x^2 \sinh (b x) \text{Shi}(b x) \, dx &=\frac{x^2 \cosh (b x) \text{Shi}(b x)}{b}-\frac{2 \int x \cosh (b x) \text{Shi}(b x) \, dx}{b}-\int \frac{x \cosh (b x) \sinh (b x)}{b} \, dx\\ &=\frac{x^2 \cosh (b x) \text{Shi}(b x)}{b}-\frac{2 x \sinh (b x) \text{Shi}(b x)}{b^2}+\frac{2 \int \sinh (b x) \text{Shi}(b x) \, dx}{b^2}-\frac{\int x \cosh (b x) \sinh (b x) \, dx}{b}+\frac{2 \int \frac{\sinh ^2(b x)}{b} \, dx}{b}\\ &=-\frac{x \sinh ^2(b x)}{2 b^2}+\frac{2 \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Shi}(b x)}{b}-\frac{2 x \sinh (b x) \text{Shi}(b x)}{b^2}+\frac{\int \sinh ^2(b x) \, dx}{2 b^2}-\frac{2 \int \frac{\cosh (b x) \sinh (b x)}{b x} \, dx}{b^2}+\frac{2 \int \sinh ^2(b x) \, dx}{b^2}\\ &=\frac{5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac{x \sinh ^2(b x)}{2 b^2}+\frac{2 \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Shi}(b x)}{b}-\frac{2 x \sinh (b x) \text{Shi}(b x)}{b^2}-\frac{2 \int \frac{\cosh (b x) \sinh (b x)}{x} \, dx}{b^3}-\frac{\int 1 \, dx}{4 b^2}-\frac{\int 1 \, dx}{b^2}\\ &=-\frac{5 x}{4 b^2}+\frac{5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac{x \sinh ^2(b x)}{2 b^2}+\frac{2 \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Shi}(b x)}{b}-\frac{2 x \sinh (b x) \text{Shi}(b x)}{b^2}-\frac{2 \int \frac{\sinh (2 b x)}{2 x} \, dx}{b^3}\\ &=-\frac{5 x}{4 b^2}+\frac{5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac{x \sinh ^2(b x)}{2 b^2}+\frac{2 \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Shi}(b x)}{b}-\frac{2 x \sinh (b x) \text{Shi}(b x)}{b^2}-\frac{\int \frac{\sinh (2 b x)}{x} \, dx}{b^3}\\ &=-\frac{5 x}{4 b^2}+\frac{5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac{x \sinh ^2(b x)}{2 b^2}+\frac{2 \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Shi}(b x)}{b}-\frac{2 x \sinh (b x) \text{Shi}(b x)}{b^2}-\frac{\text{Shi}(2 b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0763549, size = 64, normalized size = 0.71 \[ \frac{8 \text{Shi}(b x) \left (\left (b^2 x^2+2\right ) \cosh (b x)-2 b x \sinh (b x)\right )-8 \text{Shi}(2 b x)-8 b x+5 \sinh (2 b x)-2 b x \cosh (2 b x)}{8 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 68, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{3}} \left ({\it Shi} \left ( bx \right ) \left ({b}^{2}{x}^{2}\cosh \left ( bx \right ) -2\,bx\sinh \left ( bx \right ) +2\,\cosh \left ( bx \right ) \right ) -{\frac{bx \left ( \cosh \left ( bx \right ) \right ) ^{2}}{2}}+{\frac{5\,\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{4}}-{\frac{3\,bx}{4}}-{\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^{2}{\rm Shi}\left (b x\right ) \sinh \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} \sinh \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^{2} \sinh{\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^{2}{\rm Shi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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