Optimal. Leaf size=433 \[ \frac{3 \sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{Erf}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{2048 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{2048 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{\frac{x^2}{a^2}+1}}-\frac{27 a^3 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{\frac{x^2}{a^2}+1}} \]
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Rubi [A] time = 0.547982, antiderivative size = 433, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {5684, 5682, 5675, 5663, 5779, 3312, 3307, 2180, 2204, 2205, 5717, 5699} \[ \frac{3 \sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{Erf}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{2048 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{2048 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{\frac{x^2}{a^2}+1}}-\frac{27 a^3 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{\frac{x^2}{a^2}+1}} \]
Antiderivative was successfully verified.
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Rule 5684
Rule 5682
Rule 5675
Rule 5663
Rule 5779
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5717
Rule 5699
Rubi steps
\begin{align*} \int \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2} \, dx &=\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} \left (3 a^2\right ) \int \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2} \, dx-\frac{\left (3 a \sqrt{a^2+x^2}\right ) \int 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 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{\left (9 a \sqrt{a^2+x^2}\right ) \int 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{\left (1+\frac{x^2}{a^2}\right )^{3/2}}{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{64 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a^2 \sqrt{a^2+x^2}\right ) \int \frac{\sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{1+\frac{x^2}{a^2}}} \, dx}{8 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (9 \sqrt{a^2+x^2}\right ) \int \frac{x^2}{\sqrt{1+\frac{x^2}{a^2}} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{64 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cosh (2 x)}{2 \sqrt{x}}+\frac{\cosh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (9 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 )}{64 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{9 a^3 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1+\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{512 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{128 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (9 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{27 a^3 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1+\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 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 )}{1024 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 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 )}{1024 \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 )}{256 \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 )}{256 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (9 a^3 \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{128 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{27 a^3 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1+\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 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 )}{512 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 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 )}{512 \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 )}{128 \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 )}{128 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (9 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 )}{256 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (9 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 )}{256 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{27 a^3 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1+\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 a^3 \sqrt{\pi } \sqrt{a^2+x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{2048 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 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 )}{256 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 a^3 \sqrt{\pi } \sqrt{a^2+x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{2048 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 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 )}{256 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (9 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 )}{128 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (9 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 )}{128 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{27 a^3 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1+\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 \left (a^2+x^2\right )^{5/2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2+x^2\right )^{3/2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 a^3 \sqrt{\pi } \sqrt{a^2+x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{2048 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 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{3 a^3 \sqrt{\pi } \sqrt{a^2+x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{2048 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 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}}}\\ \end{align*}
Mathematica [A] time = 0.300583, size = 210, normalized size = 0.48 \[ \frac{a^3 \sqrt{a^2+x^2} \left (-5 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{5}{2},4 \sinh ^{-1}\left (\frac{x}{a}\right )\right )+5 \sqrt{-\sinh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{5}{2},-4 \sinh ^{-1}\left (\frac{x}{a}\right )\right )+60 \sqrt{2 \pi } \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )+60 \sqrt{2 \pi } \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )+384 \sinh ^{-1}\left (\frac{x}{a}\right )^3+640 \sinh \left (2 \sinh ^{-1}\left (\frac{x}{a}\right )\right ) \sinh ^{-1}\left (\frac{x}{a}\right )^2-480 \sinh ^{-1}\left (\frac{x}{a}\right ) \cosh \left (2 \sinh ^{-1}\left (\frac{x}{a}\right )\right )\right )}{2560 \sqrt{\frac{x^2}{a^2}+1} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.157, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}+{x}^{2} \right ) ^{{\frac{3}{2}}} \left ({\it Arcsinh} \left ({\frac{x}{a}} \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{\left (a^{2} + x^{2}\right )}^{\frac{3}{2}} \operatorname{arsinh}\left (\frac{x}{a}\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(-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}} \operatorname{arsinh}\left (\frac{x}{a}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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