Integrand size = 11, antiderivative size = 29 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^3} \, dx=\frac {\tan (x)}{a^3}+\frac {2 \tan ^3(x)}{3 a^3}+\frac {\tan ^5(x)}{5 a^3} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3254, 3852} \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^3} \, dx=\frac {\tan ^5(x)}{5 a^3}+\frac {2 \tan ^3(x)}{3 a^3}+\frac {\tan (x)}{a^3} \]
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Rule 3254
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^6(x) \, dx}{a^3} \\ & = -\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )}{a^3} \\ & = \frac {\tan (x)}{a^3}+\frac {2 \tan ^3(x)}{3 a^3}+\frac {\tan ^5(x)}{5 a^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^3} \, dx=\frac {\tan (x)+\frac {2 \tan ^3(x)}{3}+\frac {\tan ^5(x)}{5}}{a^3} \]
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Time = 0.58 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {\frac {\left (\tan ^{5}\left (x \right )\right )}{5}+\frac {2 \left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right )}{a^{3}}\) | \(20\) |
parallelrisch | \(\frac {\tan \left (x \right ) \left (3 \left (\sec ^{4}\left (x \right )\right )+4 \left (\sec ^{2}\left (x \right )\right )+8\right )}{15 a^{3}}\) | \(22\) |
risch | \(\frac {16 i \left (10 \,{\mathrm e}^{4 i x}+5 \,{\mathrm e}^{2 i x}+1\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5} a^{3}}\) | \(32\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {8 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {116 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{15 a}+\frac {8 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {2 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}}{a^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{5}}\) | \(69\) |
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none
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^3} \, dx=\frac {{\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}{15 \, a^{3} \cos \left (x\right )^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (27) = 54\).
Time = 1.81 (sec) , antiderivative size = 362, normalized size of antiderivative = 12.48 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^3} \, dx=- \frac {30 \tan ^{9}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{10}{\left (\frac {x}{2} \right )} - 75 a^{3} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{6}{\left (\frac {x}{2} \right )} - 150 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} - 15 a^{3}} + \frac {40 \tan ^{7}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{10}{\left (\frac {x}{2} \right )} - 75 a^{3} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{6}{\left (\frac {x}{2} \right )} - 150 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} - 15 a^{3}} - \frac {116 \tan ^{5}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{10}{\left (\frac {x}{2} \right )} - 75 a^{3} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{6}{\left (\frac {x}{2} \right )} - 150 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} - 15 a^{3}} + \frac {40 \tan ^{3}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{10}{\left (\frac {x}{2} \right )} - 75 a^{3} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{6}{\left (\frac {x}{2} \right )} - 150 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} - 15 a^{3}} - \frac {30 \tan {\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{10}{\left (\frac {x}{2} \right )} - 75 a^{3} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{6}{\left (\frac {x}{2} \right )} - 150 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} - 15 a^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^3} \, dx=\frac {3 \, \tan \left (x\right )^{5} + 10 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^3} \, dx=\frac {3 \, \tan \left (x\right )^{5} + 10 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a^{3}} \]
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Time = 12.89 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^3} \, dx=\frac {\mathrm {tan}\left (x\right )\,\left (3\,{\mathrm {tan}\left (x\right )}^4+10\,{\mathrm {tan}\left (x\right )}^2+15\right )}{15\,a^3} \]
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