Optimal. Leaf size=34 \[ -\frac{\text{Chi}(2 b x)}{2 b}+\frac{\text{Chi}(b x) \cosh (b x)}{b}-\frac{\log (x)}{2 b} \]
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Rubi [A] time = 0.0590703, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6547, 12, 3312, 3301} \[ -\frac{\text{Chi}(2 b x)}{2 b}+\frac{\text{Chi}(b x) \cosh (b x)}{b}-\frac{\log (x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6547
Rule 12
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \text{Chi}(b x) \sinh (b x) \, dx &=\frac{\cosh (b x) \text{Chi}(b x)}{b}-\int \frac{\cosh ^2(b x)}{b x} \, dx\\ &=\frac{\cosh (b x) \text{Chi}(b x)}{b}-\frac{\int \frac{\cosh ^2(b x)}{x} \, dx}{b}\\ &=\frac{\cosh (b x) \text{Chi}(b x)}{b}-\frac{\int \left (\frac{1}{2 x}+\frac{\cosh (2 b x)}{2 x}\right ) \, dx}{b}\\ &=\frac{\cosh (b x) \text{Chi}(b x)}{b}-\frac{\log (x)}{2 b}-\frac{\int \frac{\cosh (2 b x)}{x} \, dx}{2 b}\\ &=\frac{\cosh (b x) \text{Chi}(b x)}{b}-\frac{\text{Chi}(2 b x)}{2 b}-\frac{\log (x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0113208, size = 36, normalized size = 1.06 \[ -\frac{\text{Chi}(2 b x)}{2 b}+\frac{\text{Chi}(b x) \cosh (b x)}{b}-\frac{\log (b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 33, normalized size = 1. \begin{align*}{\frac{{\it Chi} \left ( bx \right ) \cosh \left ( bx \right ) }{b}}-{\frac{\ln \left ( bx \right ) }{2\,b}}-{\frac{{\it Chi} \left ( 2\,bx \right ) }{2\,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 Chi}\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 (\operatorname{Chi}\left (b x\right ) \sinh \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 \sinh{\left (b x \right )} \operatorname{Chi}\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 Chi}\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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