Optimal. Leaf size=81 \[ -\frac{3 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{2 a}+\frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a}+\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a}+x \sinh ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.118619, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {5653, 5717, 5657, 3307, 2180, 2204, 2205} \[ -\frac{3 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{2 a}+\frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a}+\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a}+x \sinh ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 5653
Rule 5717
Rule 5657
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sinh ^{-1}(a x)^{3/2} \, dx &=x \sinh ^{-1}(a x)^{3/2}-\frac{1}{2} (3 a) \int \frac{x \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{2 a}+x \sinh ^{-1}(a x)^{3/2}+\frac{3}{4} \int \frac{1}{\sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=-\frac{3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{2 a}+x \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a}\\ &=-\frac{3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{2 a}+x \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a}+\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a}\\ &=-\frac{3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{2 a}+x \sinh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a}+\frac{3 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a}\\ &=-\frac{3 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{2 a}+x \sinh ^{-1}(a x)^{3/2}+\frac{3 \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a}+\frac{3 \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.0282826, size = 47, normalized size = 0.58 \[ \frac{\frac{\sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-\sinh ^{-1}(a x)\right )}{\sqrt{\sinh ^{-1}(a x)}}-\text{Gamma}\left (\frac{5}{2},\sinh ^{-1}(a x)\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.074, size = 65, normalized size = 0.8 \begin{align*}{\frac{1}{8\,\sqrt{\pi }a} \left ( 8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3/2}\sqrt{\pi }xa-12\,\sqrt{{\it Arcsinh} \left ( ax \right ) }\sqrt{\pi }\sqrt{{a}^{2}{x}^{2}+1}+3\,\pi \,{\it Erf} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) +3\,\pi \,{\it erfi} \left ( \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 \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 \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 \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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