Optimal. Leaf size=27 \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^3}-\frac{\log \left (\sinh ^{-1}(a x)\right )}{2 a^3} \]
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Rubi [A] time = 0.144956, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5779, 3312, 3301} \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^3}-\frac{\log \left (\sinh ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 5779
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\log \left (\sinh ^{-1}(a x)\right )}{2 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^3}-\frac{\log \left (\sinh ^{-1}(a x)\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0682203, size = 22, normalized size = 0.81 \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )-\log \left (\sinh ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 24, normalized size = 0.9 \begin{align*}{\frac{{\it Chi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{2\,{a}^{3}}}-{\frac{\ln \left ({\it Arcsinh} \left ( ax \right ) \right ) }{2\,{a}^{3}}} \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^{2}}{\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^{2}}{\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^{2}}{\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^{2}}{\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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