Optimal. Leaf size=42 \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{a^2-x^2}} \]
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Rubi [A] time = 0.0613725, 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} \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{a^2-x^2}} \]
Antiderivative was successfully verified.
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Rule 4643
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{a^2-x^2}} \, dx &=\frac{\sqrt{1-\frac{x^2}{a^2}} \int \frac{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\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 )^{3/2}}{3 \sqrt{a^2-x^2}}\\ \end{align*}
Mathematica [A] time = 0.0322779, size = 42, normalized size = 1. \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{a^2-x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 38, normalized size = 0.9 \begin{align*}{\frac{2\,a}{3} \left ( \arcsin \left ({\frac{x}{a}} \right ) \right ) ^{{\frac{3}{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{\sqrt{\arcsin \left (\frac{x}{a}\right )}}{\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 = 2.279, size = 90, normalized size = 2.14 \begin{align*} -\frac{2}{3} \, \sqrt{-\arctan \left (-\frac{x}{\sqrt{a^{2} - x^{2}}}\right )} \arctan \left (-\frac{x}{\sqrt{a^{2} - x^{2}}}\right ) \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{\sqrt{\operatorname{asin}{\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 = 1.22491, size = 20, normalized size = 0.48 \begin{align*} \frac{2 \,{\left | a \right |} \arcsin \left (\frac{x}{a}\right )^{\frac{3}{2}}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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