Optimal. Leaf size=42 \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]
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Rubi [A] time = 0.0654377, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]
Antiderivative was successfully verified.
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Rule 4643
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{a^2-x^2}} \, dx &=\frac{\sqrt{1-\frac{x^2}{a^2}} \int \frac{\sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx}{\sqrt{a^2-x^2}}\\ &=\frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}}\\ \end{align*}
Mathematica [A] time = 0.0397304, size = 42, normalized size = 1. \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 38, normalized size = 0.9 \begin{align*}{\frac{2\,a}{5} \left ( \arcsin \left ({\frac{x}{a}} \right ) \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{{a}^{2}-{x}^{2}}{{a}^{2}}}}{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \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{\arcsin \left (\frac{x}{a}\right )^{\frac{3}{2}}}{\sqrt{a^{2} - x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99734, size = 92, normalized size = 2.19 \begin{align*} \frac{2}{5} \, \sqrt{-\arctan \left (-\frac{x}{\sqrt{a^{2} - x^{2}}}\right )} \arctan \left (-\frac{x}{\sqrt{a^{2} - x^{2}}}\right )^{2} \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{\operatorname{asin}^{\frac{3}{2}}{\left (\frac{x}{a} \right )}}{\sqrt{- \left (- a + x\right ) \left (a + x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26984, size = 16, normalized size = 0.38 \begin{align*} \frac{2}{5} \, \arcsin \left (\frac{x}{a}\right )^{\frac{5}{2}} \mathrm{sgn}\left (a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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