Optimal. Leaf size=226 \[ -\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \]
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Rubi [A] time = 0.236683, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4649, 4647, 4641, 4635, 4406, 12, 3305, 3351, 4723} \[ -\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \]
Antiderivative was successfully verified.
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Rule 4649
Rule 4647
Rule 4641
Rule 4635
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rule 4723
Rubi steps
\begin{align*} \int \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \, dx &=\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} \left (3 a^2\right ) \int \sqrt{a^2-x^2} \sqrt{\sin ^{-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{\sin ^{-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{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}-\frac{\left (3 a \sqrt{a^2-x^2}\right ) \int \frac{x}{\sqrt{\sin ^{-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{\sin ^{-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{\cos ^3(x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-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{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-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{\sin (2 x)}{4 \sqrt{x}}+\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-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{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-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{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-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{\sin (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-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{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-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{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\sin ^{-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{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-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 \sin \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-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 \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-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{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-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} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-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{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-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} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}\\ \end{align*}
Mathematica [C] time = 0.200631, size = 183, normalized size = 0.81 \[ \frac{a^3 \sqrt{a^2-x^2} \left (8 \sqrt{2} \sqrt{-i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-2 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+8 \sqrt{2} \sqrt{i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},2 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+\sqrt{-i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-4 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+\sqrt{i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},4 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+32 \sin ^{-1}\left (\frac{x}{a}\right )^2\right )}{128 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.256, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}-{x}^{2} \right ) ^{{\frac{3}{2}}}\sqrt{\arcsin \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{\arcsin \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{\arcsin \left (\frac{x}{a}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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