Optimal. Leaf size=81 \[ -\frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^3}+\frac{9 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{8 a^3}-\frac{x^2 \sqrt{a^2 x^2+1}}{2 a \sinh ^{-1}(a x)^2}-\frac{x}{a^2 \sinh ^{-1}(a x)}-\frac{3 x^3}{2 \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.252573, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5667, 5774, 5669, 5448, 3301, 5657} \[ -\frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^3}+\frac{9 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{8 a^3}-\frac{x^2 \sqrt{a^2 x^2+1}}{2 a \sinh ^{-1}(a x)^2}-\frac{x}{a^2 \sinh ^{-1}(a x)}-\frac{3 x^3}{2 \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5669
Rule 5448
Rule 3301
Rule 5657
Rubi steps
\begin{align*} \int \frac{x^2}{\sinh ^{-1}(a x)^3} \, dx &=-\frac{x^2 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}+\frac{\int \frac{x}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2} \, dx}{a}+\frac{1}{2} (3 a) \int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2} \, dx\\ &=-\frac{x^2 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{x}{a^2 \sinh ^{-1}(a x)}-\frac{3 x^3}{2 \sinh ^{-1}(a x)}+\frac{9}{2} \int \frac{x^2}{\sinh ^{-1}(a x)} \, dx+\frac{\int \frac{1}{\sinh ^{-1}(a x)} \, dx}{a^2}\\ &=-\frac{x^2 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{x}{a^2 \sinh ^{-1}(a x)}-\frac{3 x^3}{2 \sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac{x^2 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{x}{a^2 \sinh ^{-1}(a x)}-\frac{3 x^3}{2 \sinh ^{-1}(a x)}+\frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{a^3}+\frac{9 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 x}+\frac{\cosh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac{x^2 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{x}{a^2 \sinh ^{-1}(a x)}-\frac{3 x^3}{2 \sinh ^{-1}(a x)}+\frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{a^3}-\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac{x^2 \sqrt{1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac{x}{a^2 \sinh ^{-1}(a x)}-\frac{3 x^3}{2 \sinh ^{-1}(a x)}-\frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^3}+\frac{9 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{8 a^3}\\ \end{align*}
Mathematica [A] time = 0.136761, size = 64, normalized size = 0.79 \[ -\frac{\frac{4 a x \left (a x \sqrt{a^2 x^2+1}+\left (3 a^2 x^2+2\right ) \sinh ^{-1}(a x)\right )}{\sinh ^{-1}(a x)^2}+\text{Chi}\left (\sinh ^{-1}(a x)\right )-9 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 81, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{1}{8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{ax}{8\,{\it Arcsinh} \left ( ax \right ) }}-{\frac{{\it Chi} \left ({\it Arcsinh} \left ( ax \right ) \right ) }{8}}-{\frac{\cosh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}-{\frac{3\,\sinh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{8\,{\it Arcsinh} \left ( ax \right ) }}+{\frac{9\,{\it Chi} \left ( 3\,{\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*} \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^{2}}{\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^{2}}{\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^{2}}{\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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