Optimal. Leaf size=105 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3} \]
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Rubi [A] time = 0.142904, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5669, 5448, 3307, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 5669
Rule 5448
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{\sinh ^{-1}(a x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{x}}+\frac{\cosh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}+\frac{\operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}+\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}\\ &=-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{8 a^3}\\ \end{align*}
Mathematica [A] time = 0.0716465, size = 99, normalized size = 0.94 \[ \frac{\frac{\sqrt{3} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 \sinh ^{-1}(a x)\right )}{\sqrt{\sinh ^{-1}(a x)}}+\frac{3 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )}{\sqrt{-\sinh ^{-1}(a x)}}+3 \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )-\sqrt{3} \text{Gamma}\left (\frac{1}{2},3 \sinh ^{-1}(a x)\right )}{24 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.118, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\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 \frac{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 \frac{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 \frac{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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