Optimal. Leaf size=42 \[ \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \text {ArcSin}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {4737}
\begin {gather*} \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \text {ArcSin}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4737
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 = 0.03, size = 42, normalized size = 1.00 \begin {gather*} \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \text {ArcSin}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, 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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.09, size = 38, normalized size = 0.90 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {\operatorname {asin}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\sqrt {- \left (- a + x\right ) \left (a + x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 15, normalized size = 0.36 \begin {gather*} \frac {2 \, {\left | a \right |} \arcsin \left (\frac {x}{a}\right )^{\frac {5}{2}}}{5 \, a} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}}{\sqrt {a^2-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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