Optimal. Leaf size=82 \[ -\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^4}+\frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{a^4}-\frac{x^3 \sqrt{a^2 x^2+1}}{2 a \sinh ^{-1}(a x)^2}-\frac{3 x^2}{2 a^2 \sinh ^{-1}(a x)}-\frac{2 x^4}{\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.304009, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5667, 5774, 5669, 5448, 3298, 12} \[ -\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^4}+\frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{a^4}-\frac{x^3 \sqrt{a^2 x^2+1}}{2 a \sinh ^{-1}(a x)^2}-\frac{3 x^2}{2 a^2 \sinh ^{-1}(a x)}-\frac{2 x^4}{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5669
Rule 5448
Rule 3298
Rule 12
Rubi steps
\begin{align*} \int \frac{x^3}{\sinh ^{-1}(a x)^3} \, dx &=-\frac{x^3 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}+\frac{3 \int \frac{x^2}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2} \, dx}{2 a}+(2 a) \int \frac{x^4}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2} \, dx\\ &=-\frac{x^3 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{3 x^2}{2 a^2 \sinh ^{-1}(a x)}-\frac{2 x^4}{\sinh ^{-1}(a x)}+8 \int \frac{x^3}{\sinh ^{-1}(a x)} \, dx+\frac{3 \int \frac{x}{\sinh ^{-1}(a x)} \, dx}{a^2}\\ &=-\frac{x^3 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{3 x^2}{2 a^2 \sinh ^{-1}(a x)}-\frac{2 x^4}{\sinh ^{-1}(a x)}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=-\frac{x^3 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{3 x^2}{2 a^2 \sinh ^{-1}(a x)}-\frac{2 x^4}{\sinh ^{-1}(a x)}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac{8 \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{x^3 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{3 x^2}{2 a^2 \sinh ^{-1}(a x)}-\frac{2 x^4}{\sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}-\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=-\frac{x^3 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{3 x^2}{2 a^2 \sinh ^{-1}(a x)}-\frac{2 x^4}{\sinh ^{-1}(a x)}-\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^4}+\frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{a^4}\\ \end{align*}
Mathematica [A] time = 0.203097, size = 69, normalized size = 0.84 \[ -\frac{\frac{a^2 x^2 \left (a x \sqrt{a^2 x^2+1}+\left (4 a^2 x^2+3\right ) \sinh ^{-1}(a x)\right )}{\sinh ^{-1}(a x)^2}+\text{Shi}\left (2 \sinh ^{-1}(a x)\right )-2 \text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 82, normalized size = 1. \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{\sinh \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}+{\frac{\cosh \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{4\,{\it Arcsinh} \left ( ax \right ) }}-{\frac{{\it Shi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{2}}-{\frac{\sinh \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{16\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\cosh \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{4\,{\it Arcsinh} \left ( ax \right ) }}+{\it Shi} \left ( 4\,{\it Arcsinh} \left ( ax \right ) \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*} \text{result too large to display} \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 )^{3}}, 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}^{3}{\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 )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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