Optimal. Leaf size=71 \[ -\frac{a^2 \text{Shi}(a+b x)}{2 b^2}+\frac{\sinh (a+b x)}{2 b^2}+\frac{a \cosh (a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{Shi}(a+b x)-\frac{x \cosh (a+b x)}{2 b} \]
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Rubi [A] time = 0.211195, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6532, 6742, 2638, 3296, 2637, 3298} \[ -\frac{a^2 \text{Shi}(a+b x)}{2 b^2}+\frac{\sinh (a+b x)}{2 b^2}+\frac{a \cosh (a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{Shi}(a+b x)-\frac{x \cosh (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6532
Rule 6742
Rule 2638
Rule 3296
Rule 2637
Rule 3298
Rubi steps
\begin{align*} \int x \text{Shi}(a+b x) \, dx &=\frac{1}{2} x^2 \text{Shi}(a+b x)-\frac{1}{2} b \int \frac{x^2 \sinh (a+b x)}{a+b x} \, dx\\ &=\frac{1}{2} x^2 \text{Shi}(a+b x)-\frac{1}{2} b \int \left (-\frac{a \sinh (a+b x)}{b^2}+\frac{x \sinh (a+b x)}{b}+\frac{a^2 \sinh (a+b x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{1}{2} x^2 \text{Shi}(a+b x)-\frac{1}{2} \int x \sinh (a+b x) \, dx+\frac{a \int \sinh (a+b x) \, dx}{2 b}-\frac{a^2 \int \frac{\sinh (a+b x)}{a+b x} \, dx}{2 b}\\ &=\frac{a \cosh (a+b x)}{2 b^2}-\frac{x \cosh (a+b x)}{2 b}-\frac{a^2 \text{Shi}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{Shi}(a+b x)+\frac{\int \cosh (a+b x) \, dx}{2 b}\\ &=\frac{a \cosh (a+b x)}{2 b^2}-\frac{x \cosh (a+b x)}{2 b}+\frac{\sinh (a+b x)}{2 b^2}-\frac{a^2 \text{Shi}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{Shi}(a+b x)\\ \end{align*}
Mathematica [A] time = 0.0997287, size = 47, normalized size = 0.66 \[ \frac{\left (b^2 x^2-a^2\right ) \text{Shi}(a+b x)+\sinh (a+b x)+(a-b x) \cosh (a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 60, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{2}} \left ({\it Shi} \left ( bx+a \right ) \left ({\frac{ \left ( bx+a \right ) ^{2}}{2}}-a \left ( bx+a \right ) \right ) -{\frac{ \left ( bx+a \right ) \cosh \left ( bx+a \right ) }{2}}+{\frac{\sinh \left ( bx+a \right ) }{2}}+\cosh \left ( bx+a \right ) a \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{\rm Shi}\left (b x + a\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 \operatorname{Shi}\left (b x + a\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 \operatorname{Shi}{\left (a + 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{\rm Shi}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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