Optimal. Leaf size=34 \[ -\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3203, 3207, 2638} \[ -\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3203
Rule 3207
Rubi steps
\begin {align*} \int \left (a \sin ^2(x)\right )^{3/2} \, dx &=-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}+\frac {1}{3} (2 a) \int \sqrt {a \sin ^2(x)} \, dx\\ &=-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}+\frac {1}{3} \left (2 a \csc (x) \sqrt {a \sin ^2(x)}\right ) \int \sin (x) \, dx\\ &=-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)}-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 26, normalized size = 0.76 \[ \frac {1}{12} a (\cos (3 x)-9 \cos (x)) \csc (x) \sqrt {a \sin ^2(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 29, normalized size = 0.85 \[ \frac {{\left (a \cos \relax (x)^{3} - 3 \, a \cos \relax (x)\right )} \sqrt {-a \cos \relax (x)^{2} + a}}{3 \, \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 24, normalized size = 0.71 \[ \frac {1}{3} \, {\left ({\left (\cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) + 2 \, \mathrm {sgn}\left (\sin \relax (x)\right )\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.82, size = 24, normalized size = 0.71 \[ -\frac {a^{2} \sin \relax (x ) \cos \relax (x ) \left (2+\sin ^{2}\relax (x )\right )}{3 \sqrt {a \left (\sin ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \relax (x)^{2}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a\,{\sin \relax (x)}^2\right )}^{3/2} \,d x \]
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 \[ \int \left (a \sin ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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