Optimal. Leaf size=9 \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a^2} \]
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Rubi [A] time = 0.0801654, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5779, 3298} \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 5779
Rule 3298
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=\frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0258395, size = 9, normalized size = 1. \[ \frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 10, normalized size = 1.1 \begin{align*}{\frac{{\it Shi} \left ({\it Arcsinh} \left ( ax \right ) \right ) }{{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}{\sqrt{a^{2} x^{2} + 1} \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}{\sqrt{a^{2} x^{2} + 1} \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}{\sqrt{a^{2} x^{2} + 1} \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}{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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