Optimal. Leaf size=3 \[ \text{Shi}\left (\sinh ^{-1}(x)\right ) \]
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Rubi [A] time = 0.0595927, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5779, 3298} \[ \text{Shi}\left (\sinh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Rule 5779
Rule 3298
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{1+x^2} \sinh ^{-1}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(x)\right )\\ &=\text{Shi}\left (\sinh ^{-1}(x)\right )\\ \end{align*}
Mathematica [A] time = 0.0583648, size = 3, normalized size = 1. \[ \text{Shi}\left (\sinh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 4, normalized size = 1.3 \begin{align*}{\it Shi} \left ({\it Arcsinh} \left ( x \right ) \right ) \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{x^{2} + 1} \operatorname{arsinh}\left (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{x^{2} + 1} \operatorname{arsinh}\left (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{x^{2} + 1} \operatorname{asinh}{\left (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{x^{2} + 1} \operatorname{arsinh}\left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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