Optimal. Leaf size=130 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}+\frac{\sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{2 x^2 \sqrt{a^2 x^2+1}}{a \sqrt{\sinh ^{-1}(a x)}} \]
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Rubi [A] time = 0.122016, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5665, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}+\frac{\sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{2 x^2 \sqrt{a^2 x^2+1}}{a \sqrt{\sinh ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 5665
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x^2}{\sinh ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 \sqrt{x}}+\frac{3 \sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2 a^3}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2 a^3}-\frac{3 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2 a^3}+\frac{3 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2 a^3}\\ &=-\frac{2 x^2 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}+\frac{\sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{4 a^3}\\ \end{align*}
Mathematica [A] time = 0.13127, size = 140, normalized size = 1.08 \[ \frac{\sqrt{3} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 \sinh ^{-1}(a x)\right )-\sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )-\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )+\sqrt{3} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 \sinh ^{-1}(a x)\right )-e^{-3 \sinh ^{-1}(a x)}+e^{-\sinh ^{-1}(a x)}+e^{\sinh ^{-1}(a x)}-e^{3 \sinh ^{-1}(a x)}}{4 a^3 \sqrt{\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, 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}}{\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^{2}}{\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^{2}}{\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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