Integrand size = 11, antiderivative size = 51 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {\tan (x)}{a^5}+\frac {4 \tan ^3(x)}{3 a^5}+\frac {6 \tan ^5(x)}{5 a^5}+\frac {4 \tan ^7(x)}{7 a^5}+\frac {\tan ^9(x)}{9 a^5} \]
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Time = 0.02 (sec) , antiderivative size = 51, 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} \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {\tan ^9(x)}{9 a^5}+\frac {4 \tan ^7(x)}{7 a^5}+\frac {6 \tan ^5(x)}{5 a^5}+\frac {4 \tan ^3(x)}{3 a^5}+\frac {\tan (x)}{a^5} \]
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Rule 3254
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^{10}(x) \, dx}{a^5} \\ & = -\frac {\text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-\tan (x)\right )}{a^5} \\ & = \frac {\tan (x)}{a^5}+\frac {4 \tan ^3(x)}{3 a^5}+\frac {6 \tan ^5(x)}{5 a^5}+\frac {4 \tan ^7(x)}{7 a^5}+\frac {\tan ^9(x)}{9 a^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {\tan (x)+\frac {4 \tan ^3(x)}{3}+\frac {6 \tan ^5(x)}{5}+\frac {4 \tan ^7(x)}{7}+\frac {\tan ^9(x)}{9}}{a^5} \]
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Time = 1.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {\frac {\left (\tan ^{9}\left (x \right )\right )}{9}+\frac {4 \left (\tan ^{7}\left (x \right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (x \right )\right )}{5}+\frac {4 \left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right )}{a^{5}}\) | \(32\) |
parallelrisch | \(\frac {\tan \left (x \right ) \left (\sec ^{8}\left (x \right )\right ) \left (128+\cos \left (8 x \right )+10 \cos \left (6 x \right )+46 \cos \left (4 x \right )+130 \cos \left (2 x \right )\right )}{315 a^{5}}\) | \(36\) |
risch | \(\frac {256 i \left (126 \,{\mathrm e}^{8 i x}+84 \,{\mathrm e}^{6 i x}+36 \,{\mathrm e}^{4 i x}+9 \,{\mathrm e}^{2 i x}+1\right )}{315 \left ({\mathrm e}^{2 i x}+1\right )^{9} a^{5}}\) | \(46\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {{\left (128 \, \cos \left (x\right )^{8} + 64 \, \cos \left (x\right )^{6} + 48 \, \cos \left (x\right )^{4} + 40 \, \cos \left (x\right )^{2} + 35\right )} \sin \left (x\right )}{315 \, a^{5} \cos \left (x\right )^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (51) = 102\).
Time = 14.84 (sec) , antiderivative size = 1083, normalized size of antiderivative = 21.24 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\text {Too large to display} \]
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Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {35 \, \tan \left (x\right )^{9} + 180 \, \tan \left (x\right )^{7} + 378 \, \tan \left (x\right )^{5} + 420 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{315 \, a^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {35 \, \tan \left (x\right )^{9} + 180 \, \tan \left (x\right )^{7} + 378 \, \tan \left (x\right )^{5} + 420 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{315 \, a^{5}} \]
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Time = 13.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {\mathrm {tan}\left (x\right )}{a^5}+\frac {4\,{\mathrm {tan}\left (x\right )}^3}{3\,a^5}+\frac {6\,{\mathrm {tan}\left (x\right )}^5}{5\,a^5}+\frac {4\,{\mathrm {tan}\left (x\right )}^7}{7\,a^5}+\frac {{\mathrm {tan}\left (x\right )}^9}{9\,a^5} \]
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