Integrand size = 13, antiderivative size = 53 \[ \int \left (a-a \sin ^2(x)\right )^{5/2} \, dx=\frac {8}{15} a^2 \sqrt {a \cos ^2(x)} \tan (x)+\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x) \]
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Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3255, 3282, 3286, 2717} \[ \int \left (a-a \sin ^2(x)\right )^{5/2} \, dx=\frac {8}{15} a^2 \tan (x) \sqrt {a \cos ^2(x)}+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}+\frac {4}{15} a \tan (x) \left (a \cos ^2(x)\right )^{3/2} \]
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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 )^{5/2} \, dx \\ & = \frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{5} (4 a) \int \left (a \cos ^2(x)\right )^{3/2} \, dx \\ & = \frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{15} \left (8 a^2\right ) \int \sqrt {a \cos ^2(x)} \, dx \\ & = \frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{15} \left (8 a^2 \sqrt {a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx \\ & = \frac {8}{15} a^2 \sqrt {a \cos ^2(x)} \tan (x)+\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.62 \[ \int \left (a-a \sin ^2(x)\right )^{5/2} \, dx=\frac {1}{15} a^2 \sqrt {a \cos ^2(x)} \left (15-10 \sin ^2(x)+3 \sin ^4(x)\right ) \tan (x) \]
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Time = 1.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {\cos \left (x \right ) a^{3} \sin \left (x \right ) \left (3 \left (\cos ^{4}\left (x \right )\right )+4 \left (\cos ^{2}\left (x \right )\right )+8\right )}{15 \sqrt {a \left (\cos ^{2}\left (x \right )\right )}}\) | \(32\) |
risch | \(-\frac {i a^{2} {\mathrm e}^{6 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{160 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {5 i a^{2} {\mathrm e}^{2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{16 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {5 i a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{16 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {5 i a^{2} {\mathrm e}^{-2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{96 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {11 i a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, \cos \left (4 x \right )}{240 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {7 a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, \sin \left (4 x \right )}{120 \left ({\mathrm e}^{2 i x}+1\right )}\) | \(222\) |
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Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75 \[ \int \left (a-a \sin ^2(x)\right )^{5/2} \, dx=\frac {{\left (3 \, a^{2} \cos \left (x\right )^{4} + 4 \, a^{2} \cos \left (x\right )^{2} + 8 \, a^{2}\right )} \sqrt {a \cos \left (x\right )^{2}} \sin \left (x\right )}{15 \, \cos \left (x\right )} \]
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Timed out. \[ \int \left (a-a \sin ^2(x)\right )^{5/2} \, dx=\text {Timed out} \]
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Time = 0.38 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.58 \[ \int \left (a-a \sin ^2(x)\right )^{5/2} \, dx=\frac {1}{240} \, {\left (3 \, a^{2} \sin \left (5 \, x\right ) + 25 \, a^{2} \sin \left (3 \, x\right ) + 150 \, a^{2} \sin \left (x\right )\right )} \sqrt {a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (41) = 82\).
Time = 0.38 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.58 \[ \int \left (a-a \sin ^2(x)\right )^{5/2} \, dx=-\frac {2 \, {\left (15 \, a^{\frac {5}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{4} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) - 40 \, a^{\frac {5}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) + 48 \, a^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )\right )}}{15 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{5}} \]
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Timed out. \[ \int \left (a-a \sin ^2(x)\right )^{5/2} \, dx=\int {\left (a-a\,{\sin \left (x\right )}^2\right )}^{5/2} \,d x \]
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