Optimal. Leaf size=120 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.244318, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5663, 5779, 3312, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5779
Rule 3312
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^2 \sqrt{\sinh ^{-1}(a x)} \, dx &=\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)}-\frac{1}{6} a \int \frac{x^3}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)}-\frac{i \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 \sqrt{x}}-\frac{i \sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{24 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^3}+\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{24 a^3}-\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{24 a^3}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}+\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sinh ^{-1}(a x)}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{16 a^3}-\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}\\ \end{align*}
Mathematica [A] time = 0.0344497, size = 101, normalized size = 0.84 \[ \frac{\sqrt{3} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-3 \sinh ^{-1}(a x)\right )-9 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-\sinh ^{-1}(a x)\right )+\sqrt{-\sinh ^{-1}(a x)} \left (9 \text{Gamma}\left (\frac{3}{2},\sinh ^{-1}(a x)\right )-\sqrt{3} \text{Gamma}\left (\frac{3}{2},3 \sinh ^{-1}(a x)\right )\right )}{72 a^3 \sqrt{-\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.097, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sqrt{{\it Arcsinh} \left ( ax \right ) }\, dx \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 x^{2} \sqrt{\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(-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 x^{2} \sqrt{\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 x^{2} \sqrt{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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