Optimal. Leaf size=309 \[ \frac{\sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{Erf}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{\frac{x^2}{a^2}+1}}+\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \]
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Rubi [A] time = 0.360075, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5684, 5682, 5675, 5669, 5448, 12, 3308, 2180, 2204, 2205, 5779} \[ \frac{\sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{Erf}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{\frac{x^2}{a^2}+1}}+\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \]
Antiderivative was successfully verified.
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Rule 5684
Rule 5682
Rule 5675
Rule 5669
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5779
Rubi steps
\begin{align*} \int \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \, dx &=\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} \left (3 a^2\right ) \int \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \, dx-\frac{\left (a \sqrt{a^2+x^2}\right ) \int \frac{x \left (1+\frac{x^2}{a^2}\right )}{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{8 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}-\frac{\left (3 a \sqrt{a^2+x^2}\right ) \int \frac{x}{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{16 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a^2 \sqrt{a^2+x^2}\right ) \int \frac{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{1+\frac{x^2}{a^2}}} \, dx}{8 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{8 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 \sqrt{x}}+\frac{\sinh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{8 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{16 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{16 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{128 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{128 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1+\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{\pi } \sqrt{a^2+x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{1+\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2+x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\pi } \sqrt{a^2+x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{1+\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2+x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1+\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{\pi } \sqrt{a^2+x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{1+\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2+x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{1+\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\pi } \sqrt{a^2+x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{1+\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2+x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{1+\frac{x^2}{a^2}}}\\ \end{align*}
Mathematica [A] time = 0.174533, size = 156, normalized size = 0.5 \[ \frac{a^3 \sqrt{a^2+x^2} \left (-\sqrt{-\sinh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-4 \sinh ^{-1}\left (\frac{x}{a}\right )\right )-8 \sqrt{2} \sqrt{-\sinh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-2 \sinh ^{-1}\left (\frac{x}{a}\right )\right )+\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \left (-8 \sqrt{2} \text{Gamma}\left (\frac{3}{2},2 \sinh ^{-1}\left (\frac{x}{a}\right )\right )-\text{Gamma}\left (\frac{3}{2},4 \sinh ^{-1}\left (\frac{x}{a}\right )\right )+32 \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}\right )\right )}{128 \sqrt{\frac{x^2}{a^2}+1} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.198, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}+{x}^{2} \right ) ^{{\frac{3}{2}}}\sqrt{{\it Arcsinh} \left ({\frac{x}{a}} \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{\left (a^{2} + x^{2}\right )}^{\frac{3}{2}} \sqrt{\operatorname{arsinh}\left (\frac{x}{a}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (a^{2} + x^{2}\right )}^{\frac{3}{2}} \sqrt{\operatorname{arsinh}\left (\frac{x}{a}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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