Optimal. Leaf size=84 \[ \frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a^2}+\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a^2}-\frac{2 x \sqrt{a^2 x^2+1}}{a \sqrt{\sinh ^{-1}(a x)}} \]
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Rubi [A] time = 0.0663937, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5665, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a^2}+\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a^2}-\frac{2 x \sqrt{a^2 x^2+1}}{a \sqrt{\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 5665
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x}{\sinh ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{2 x \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{2 x \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{a^2}+\frac{2 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{a^2}\\ &=-\frac{2 x \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a^2}+\frac{\sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0264979, size = 78, normalized size = 0.93 \[ \frac{\sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 \sinh ^{-1}(a x)\right )}{\sqrt{2} a^2 \sqrt{\sinh ^{-1}(a x)}}-\frac{\text{Gamma}\left (\frac{1}{2},2 \sinh ^{-1}(a x)\right )}{\sqrt{2} a^2}-\frac{\sinh \left (2 \sinh ^{-1}(a x)\right )}{a^2 \sqrt{\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.086, size = 80, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{2\,\sqrt{\pi }{a}^{2}{\it Arcsinh} \left ( ax \right ) } \left ( -2\,\sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) }\sqrt{\pi }\sqrt{{a}^{2}{x}^{2}+1}xa+{\it Arcsinh} \left ( ax \right ) \pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) +{\it Arcsinh} \left ( ax \right ) \pi \,{\it erfi} \left ( \sqrt{2}\sqrt{{\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*} \int \frac{x}{\operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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}{\operatorname{asinh}^{\frac{3}{2}}{\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}{\operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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