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 = 0.07, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 4641
Rule 4643
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*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 1.00 \[ \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.60, size = 38, normalized size = 0.90 \[ \frac {2}{5} \, \sqrt {-\arctan \left (-\frac {x}{\sqrt {a^{2} - x^{2}}}\right )} \arctan \left (-\frac {x}{\sqrt {a^{2} - x^{2}}}\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 15, normalized size = 0.36 \[ \frac {2 \, {\left | a \right |} \arcsin \left (\frac {x}{a}\right )^{\frac {5}{2}}}{5 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 38, normalized size = 0.90
method | result | size |
default | \(\frac {2 \arcsin \left (\frac {x}{a}\right )^{\frac {5}{2}} a \sqrt {\frac {a^{2}-x^{2}}{a^{2}}}}{5 \sqrt {a^{2}-x^{2}}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}}{\sqrt {a^2-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\sqrt {- \left (- a + x\right ) \left (a + x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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