Optimal. Leaf size=109 \[ \frac{a \text{Chi}(2 a+2 b x)}{2 b^2}-\frac{\text{Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac{\text{Shi}(2 a+2 b x)}{2 b^2}+\frac{a \log (a+b x)}{2 b^2}-\frac{\sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac{x \text{Chi}(a+b x) \cosh (a+b x)}{b}-\frac{x}{2 b} \]
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Rubi [A] time = 0.224368, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {6549, 6742, 2635, 8, 3312, 3301, 6541, 5448, 12, 3298} \[ \frac{a \text{Chi}(2 a+2 b x)}{2 b^2}-\frac{\text{Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac{\text{Shi}(2 a+2 b x)}{2 b^2}+\frac{a \log (a+b x)}{2 b^2}-\frac{\sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac{x \text{Chi}(a+b x) \cosh (a+b x)}{b}-\frac{x}{2 b} \]
Antiderivative was successfully verified.
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Rule 6549
Rule 6742
Rule 2635
Rule 8
Rule 3312
Rule 3301
Rule 6541
Rule 5448
Rule 12
Rule 3298
Rubi steps
\begin{align*} \int x \text{Chi}(a+b x) \sinh (a+b x) \, dx &=\frac{x \cosh (a+b x) \text{Chi}(a+b x)}{b}-\frac{\int \cosh (a+b x) \text{Chi}(a+b x) \, dx}{b}-\int \frac{x \cosh ^2(a+b x)}{a+b x} \, dx\\ &=\frac{x \cosh (a+b x) \text{Chi}(a+b x)}{b}-\frac{\text{Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac{\int \frac{\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{b}-\int \left (\frac{\cosh ^2(a+b x)}{b}-\frac{a \cosh ^2(a+b x)}{b (a+b x)}\right ) \, dx\\ &=\frac{x \cosh (a+b x) \text{Chi}(a+b x)}{b}-\frac{\text{Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac{\int \cosh ^2(a+b x) \, dx}{b}+\frac{\int \frac{\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{b}+\frac{a \int \frac{\cosh ^2(a+b x)}{a+b x} \, dx}{b}\\ &=\frac{x \cosh (a+b x) \text{Chi}(a+b x)}{b}-\frac{\cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac{\text{Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac{\int 1 \, dx}{2 b}+\frac{\int \frac{\sinh (2 a+2 b x)}{a+b x} \, dx}{2 b}+\frac{a \int \left (\frac{1}{2 (a+b x)}+\frac{\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac{x}{2 b}+\frac{x \cosh (a+b x) \text{Chi}(a+b x)}{b}+\frac{a \log (a+b x)}{2 b^2}-\frac{\cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac{\text{Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac{\text{Shi}(2 a+2 b x)}{2 b^2}+\frac{a \int \frac{\cosh (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=-\frac{x}{2 b}+\frac{x \cosh (a+b x) \text{Chi}(a+b x)}{b}+\frac{a \text{Chi}(2 a+2 b x)}{2 b^2}+\frac{a \log (a+b x)}{2 b^2}-\frac{\cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac{\text{Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac{\text{Shi}(2 a+2 b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.184971, size = 78, normalized size = 0.72 \[ \frac{2 a \text{Chi}(2 (a+b x))+4 \text{Chi}(a+b x) (b x \cosh (a+b x)-\sinh (a+b x))+2 \text{Shi}(2 (a+b x))+2 a \log (a+b x)-\sinh (2 (a+b x))-2 b x}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 106, normalized size = 1. \begin{align*}{\frac{x{\it Chi} \left ( bx+a \right ) \cosh \left ( bx+a \right ) }{b}}-{\frac{{\it Chi} \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{{b}^{2}}}-{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2\,{b}^{2}}}-{\frac{x}{2\,b}}-{\frac{a}{2\,{b}^{2}}}+{\frac{{\it Shi} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{2}}}+{\frac{a\ln \left ( bx+a \right ) }{2\,{b}^{2}}}+{\frac{a{\it Chi} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{2}}} \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 Chi}\left (b x + a\right ) \sinh \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{Chi}\left (b x + a\right ) \sinh \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 \sinh{\left (a + b x \right )} \operatorname{Chi}\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 Chi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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