Optimal. Leaf size=14 \[ \frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^2} \]
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Rubi [A] time = 0.0379933, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5669, 5448, 12, 3298} \[ \frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 5669
Rule 5448
Rule 12
Rule 3298
Rubi steps
\begin{align*} \int \frac{x}{\sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^2}\\ &=\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0202518, size = 14, normalized size = 1. \[ \frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 13, normalized size = 0.9 \begin{align*}{\frac{{\it Shi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{2\,{a}^{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 \frac{x}{\operatorname{arsinh}\left (a 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 (\frac{x}{\operatorname{arsinh}\left (a 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 \frac{x}{\operatorname{asinh}{\left (a 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 \frac{x}{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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