Optimal. Leaf size=41 \[ -\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^5}+\frac{\text{Chi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^5}+\frac{3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5} \]
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Rubi [A] time = 0.161055, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5779, 3312, 3301} \[ -\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^5}+\frac{\text{Chi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^5}+\frac{3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5} \]
Antiderivative was successfully verified.
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Rule 5779
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^4(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 x}-\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac{3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^5}+\frac{\text{Chi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^5}+\frac{3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5}\\ \end{align*}
Mathematica [A] time = 0.0710375, size = 31, normalized size = 0.76 \[ \frac{-4 \text{Chi}\left (2 \sinh ^{-1}(a x)\right )+\text{Chi}\left (4 \sinh ^{-1}(a x)\right )+3 \log \left (\sinh ^{-1}(a x)\right )}{8 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 36, normalized size = 0.9 \begin{align*} -{\frac{{\it Chi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{2\,{a}^{5}}}+{\frac{{\it Chi} \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{8\,{a}^{5}}}+{\frac{3\,\ln \left ({\it Arcsinh} \left ( ax \right ) \right ) }{8\,{a}^{5}}} \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^{4}}{\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^{4}}{\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^{4}}{\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^{4}}{\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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