Optimal. Leaf size=215 \[ -\frac{3 \sqrt{\pi } a \sqrt{a^2-x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 a \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1-\frac{x^2}{a^2}}} \]
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Rubi [A] time = 0.231775, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4647, 4641, 4629, 4723, 3312, 3304, 3352} \[ -\frac{3 \sqrt{\pi } a \sqrt{a^2-x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 a \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1-\frac{x^2}{a^2}}} \]
Antiderivative was successfully verified.
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Rule 4647
Rule 4641
Rule 4629
Rule 4723
Rule 3312
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2} \, dx &=\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{\sqrt{a^2-x^2} \int \frac{\sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx}{2 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 \sqrt{a^2-x^2}\right ) \int x \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \, dx}{4 a \sqrt{1-\frac{x^2}{a^2}}}\\ &=-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}+\frac{\left (3 \sqrt{a^2-x^2}\right ) \int \frac{x^2}{\sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}} \, dx}{16 a^2 \sqrt{1-\frac{x^2}{a^2}}}\\ &=-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}+\frac{\left (3 a \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 )}{16 \sqrt{1-\frac{x^2}{a^2}}}\\ &=-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}+\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}\left (\frac{x}{a}\right )\right )}{16 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{3 a \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}\left (\frac{x}{a}\right )\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{3 a \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \cos \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 a \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 a \sqrt{\pi } \sqrt{a^2-x^2} C\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}\\ \end{align*}
Mathematica [C] time = 0.13617, size = 173, normalized size = 0.8 \[ \frac{\sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \left (15 \sqrt{2} a \sqrt{i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-2 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+15 \sqrt{2} a \sqrt{-i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},2 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+32 \sin ^{-1}\left (\frac{x}{a}\right ) \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )^2} \left (5 x \sqrt{1-\frac{x^2}{a^2}}+2 a \sin ^{-1}\left (\frac{x}{a}\right )\right )\right )}{320 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.236, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{a}^{2}-{x}^{2}} \left ( \arcsin \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 \sqrt{a^{2} - x^{2}} \arcsin \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 \sqrt{a^{2} - x^{2}} \arcsin \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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