Optimal. Leaf size=31 \[ x \text{Shi}(b x)^2+\frac{\text{Shi}(2 b x)}{b}-\frac{2 \text{Shi}(b x) \cosh (b x)}{b} \]
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Rubi [A] time = 0.0577705, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6534, 6540, 12, 5448, 3298} \[ x \text{Shi}(b x)^2+\frac{\text{Shi}(2 b x)}{b}-\frac{2 \text{Shi}(b x) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6534
Rule 6540
Rule 12
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int \text{Shi}(b x)^2 \, dx &=x \text{Shi}(b x)^2-2 \int \sinh (b x) \text{Shi}(b x) \, dx\\ &=-\frac{2 \cosh (b x) \text{Shi}(b x)}{b}+x \text{Shi}(b x)^2+2 \int \frac{\cosh (b x) \sinh (b x)}{b x} \, dx\\ &=-\frac{2 \cosh (b x) \text{Shi}(b x)}{b}+x \text{Shi}(b x)^2+\frac{2 \int \frac{\cosh (b x) \sinh (b x)}{x} \, dx}{b}\\ &=-\frac{2 \cosh (b x) \text{Shi}(b x)}{b}+x \text{Shi}(b x)^2+\frac{2 \int \frac{\sinh (2 b x)}{2 x} \, dx}{b}\\ &=-\frac{2 \cosh (b x) \text{Shi}(b x)}{b}+x \text{Shi}(b x)^2+\frac{\int \frac{\sinh (2 b x)}{x} \, dx}{b}\\ &=-\frac{2 \cosh (b x) \text{Shi}(b x)}{b}+x \text{Shi}(b x)^2+\frac{\text{Shi}(2 b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0070912, size = 31, normalized size = 1. \[ x \text{Shi}(b x)^2+\frac{\text{Shi}(2 b x)}{b}-\frac{2 \text{Shi}(b x) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 30, normalized size = 1. \begin{align*}{\frac{bx \left ({\it Shi} \left ( bx \right ) \right ) ^{2}-2\,\cosh \left ( bx \right ){\it Shi} \left ( bx \right ) +{\it Shi} \left ( 2\,bx \right ) }{b}} \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{\rm Shi}\left (b x\right )^{2}\,{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 (\operatorname{Shi}\left (b x\right )^{2}, 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 \operatorname{Shi}^{2}{\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{\rm Shi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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