Integrand size = 10, antiderivative size = 34 \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)}-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3282, 3286, 2718} \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)} \]
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Rule 2718
Rule 3282
Rule 3286
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\frac {1}{12} a (-9 \cos (x)+\cos (3 x)) \csc (x) \sqrt {a \sin ^2(x)} \]
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Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {a^{2} \sin \left (x \right ) \cos \left (x \right ) \left (\sin ^{2}\left (x \right )+2\right )}{3 \sqrt {a \left (\sin ^{2}\left (x \right )\right )}}\) | \(24\) |
risch | \(\frac {i a \,{\mathrm e}^{4 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}-24}-\frac {3 i a \,{\mathrm e}^{2 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {3 i \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, a}{8 \left ({\mathrm e}^{2 i x}-1\right )}+\frac {i a \,{\mathrm e}^{-2 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}-24}\) | \(145\) |
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\frac {{\left (a \cos \left (x\right )^{3} - 3 \, a \cos \left (x\right )\right )} \sqrt {-a \cos \left (x\right )^{2} + a}}{3 \, \sin \left (x\right )} \]
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\[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\int \left (a \sin ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\int { \left (a \sin \left (x\right )^{2}\right )^{\frac {3}{2}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\frac {1}{3} \, {\left ({\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) + 2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )\right )} a^{\frac {3}{2}} \]
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Timed out. \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\int {\left (a\,{\sin \left (x\right )}^2\right )}^{3/2} \,d x \]
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