Optimal. Leaf size=97 \[ \frac{\text{Chi}(2 a+2 b x)}{2 b^2}-\frac{\text{Chi}(a+b x) \cosh (a+b x)}{b^2}+\frac{a \text{Shi}(2 a+2 b x)}{2 b^2}+\frac{\log (a+b x)}{2 b^2}-\frac{\cosh (2 a+2 b x)}{4 b^2}+\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b} \]
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Rubi [A] time = 0.251418, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6543, 5617, 6741, 6742, 2638, 3298, 6547, 3312, 3301} \[ \frac{\text{Chi}(2 a+2 b x)}{2 b^2}-\frac{\text{Chi}(a+b x) \cosh (a+b x)}{b^2}+\frac{a \text{Shi}(2 a+2 b x)}{2 b^2}+\frac{\log (a+b x)}{2 b^2}-\frac{\cosh (2 a+2 b x)}{4 b^2}+\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6543
Rule 5617
Rule 6741
Rule 6742
Rule 2638
Rule 3298
Rule 6547
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int x \cosh (a+b x) \text{Chi}(a+b x) \, dx &=\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b}-\frac{\int \text{Chi}(a+b x) \sinh (a+b x) \, dx}{b}-\int \frac{x \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx\\ &=-\frac{\cosh (a+b x) \text{Chi}(a+b x)}{b^2}+\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b}-\frac{1}{2} \int \frac{x \sinh (2 (a+b x))}{a+b x} \, dx+\frac{\int \frac{\cosh ^2(a+b x)}{a+b x} \, dx}{b}\\ &=-\frac{\cosh (a+b x) \text{Chi}(a+b x)}{b^2}+\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b}-\frac{1}{2} \int \frac{x \sinh (2 a+2 b x)}{a+b x} \, dx+\frac{\int \left (\frac{1}{2 (a+b x)}+\frac{\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac{\cosh (a+b x) \text{Chi}(a+b x)}{b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b}-\frac{1}{2} \int \left (\frac{\sinh (2 a+2 b x)}{b}+\frac{a \sinh (2 a+2 b x)}{b (-a-b x)}\right ) \, dx+\frac{\int \frac{\cosh (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=-\frac{\cosh (a+b x) \text{Chi}(a+b x)}{b^2}+\frac{\text{Chi}(2 a+2 b x)}{2 b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b}-\frac{\int \sinh (2 a+2 b x) \, dx}{2 b}-\frac{a \int \frac{\sinh (2 a+2 b x)}{-a-b x} \, dx}{2 b}\\ &=-\frac{\cosh (2 a+2 b x)}{4 b^2}-\frac{\cosh (a+b x) \text{Chi}(a+b x)}{b^2}+\frac{\text{Chi}(2 a+2 b x)}{2 b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b}+\frac{a \text{Shi}(2 a+2 b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.160026, size = 73, normalized size = 0.75 \[ \frac{2 \text{Chi}(2 (a+b x))+4 \text{Chi}(a+b x) (b x \sinh (a+b x)-\cosh (a+b x))+2 a \text{Shi}(2 (a+b x))+2 \log (a+b x)-\cosh (2 (a+b x))}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 89, normalized size = 0.9 \begin{align*}{\frac{x{\it Chi} \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{b}}-{\frac{{\it Chi} \left ( bx+a \right ) \cosh \left ( bx+a \right ) }{{b}^{2}}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2\,{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) }{2\,{b}^{2}}}+{\frac{{\it Chi} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{2}}}+{\frac{a{\it Shi} \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 ) \cosh \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 \cosh \left (b x + a\right ) \operatorname{Chi}\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 \cosh{\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 ) \cosh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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