Integrand size = 13, antiderivative size = 34 \[ \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx=\frac {2}{3} a \sqrt {a \cos ^2(x)} \tan (x)+\frac {1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x) \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3255, 3282, 3286, 2717} \[ \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx=\frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}+\frac {2}{3} a \tan (x) \sqrt {a \cos ^2(x)} \]
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Rule 2717
Rule 3255
Rule 3282
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \int \left (a \cos ^2(x)\right )^{3/2} \, dx \\ & = \frac {1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{3} (2 a) \int \sqrt {a \cos ^2(x)} \, dx \\ & = \frac {1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{3} \left (2 a \sqrt {a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx \\ & = \frac {2}{3} a \sqrt {a \cos ^2(x)} \tan (x)+\frac {1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx=-\frac {1}{3} a \sqrt {a \cos ^2(x)} \left (-3+\sin ^2(x)\right ) \tan (x) \]
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Time = 0.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {\cos \left (x \right ) a^{2} \sin \left (x \right ) \left (\sin ^{2}\left (x \right )-3\right )}{3 \sqrt {a \left (\cos ^{2}\left (x \right )\right )}}\) | \(24\) |
risch | \(-\frac {i a \,{\mathrm e}^{4 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {3 i a \,{\mathrm e}^{2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {3 i a \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {i a \,{\mathrm e}^{-2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}+24}\) | \(141\) |
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx=\frac {{\left (a \cos \left (x\right )^{2} + 2 \, a\right )} \sqrt {a \cos \left (x\right )^{2}} \sin \left (x\right )}{3 \, \cos \left (x\right )} \]
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\[ \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx=\int \left (- a \sin ^{2}{\left (x \right )} + a\right )^{\frac {3}{2}}\, dx \]
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Time = 0.37 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.50 \[ \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx=\frac {1}{12} \, {\left (a \sin \left (3 \, x\right ) + 9 \, a \sin \left (x\right )\right )} \sqrt {a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (26) = 52\).
Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx=-\frac {2 \, {\left (3 \, a^{\frac {3}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) - 4 \, a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )\right )}}{3 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{3}} \]
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Timed out. \[ \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx=\int {\left (a-a\,{\sin \left (x\right )}^2\right )}^{3/2} \,d x \]
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