Optimal. Leaf size=29 \[ \frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4} \]
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Rubi [A] time = 0.0679608, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5669, 5448, 3298} \[ \frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4} \]
Antiderivative was successfully verified.
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Rule 5669
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int \frac{x^3}{\sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4}+\frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}\\ \end{align*}
Mathematica [A] time = 0.0717786, size = 24, normalized size = 0.83 \[ \frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )-2 \text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 24, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{4}} \left ( -{\frac{{\it Shi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{4}}+{\frac{{\it Shi} \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{8}} \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^{3}}{\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^{3}}{\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^{3}}{\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^{3}}{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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