Integrand size = 11, antiderivative size = 37 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {\tan (x)}{a^4}+\frac {\tan ^3(x)}{a^4}+\frac {3 \tan ^5(x)}{5 a^4}+\frac {\tan ^7(x)}{7 a^4} \]
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Time = 0.02 (sec) , antiderivative size = 37, 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} \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {\tan ^7(x)}{7 a^4}+\frac {3 \tan ^5(x)}{5 a^4}+\frac {\tan ^3(x)}{a^4}+\frac {\tan (x)}{a^4} \]
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Rule 3254
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^8(x) \, dx}{a^4} \\ & = -\frac {\text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (x)\right )}{a^4} \\ & = \frac {\tan (x)}{a^4}+\frac {\tan ^3(x)}{a^4}+\frac {3 \tan ^5(x)}{5 a^4}+\frac {\tan ^7(x)}{7 a^4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {\tan (x)+\tan ^3(x)+\frac {3 \tan ^5(x)}{5}+\frac {\tan ^7(x)}{7}}{a^4} \]
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Time = 0.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65
method | result | size |
default | \(\frac {\frac {\left (\tan ^{7}\left (x \right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (x \right )\right )}{5}+\tan ^{3}\left (x \right )+\tan \left (x \right )}{a^{4}}\) | \(24\) |
parallelrisch | \(\frac {\tan \left (x \right ) \left (\sec ^{6}\left (x \right )\right ) \left (32+\cos \left (6 x \right )+8 \cos \left (4 x \right )+29 \cos \left (2 x \right )\right )}{70 a^{4}}\) | \(30\) |
risch | \(\frac {32 i \left (35 \,{\mathrm e}^{6 i x}+21 \,{\mathrm e}^{4 i x}+7 \,{\mathrm e}^{2 i x}+1\right )}{35 \left ({\mathrm e}^{2 i x}+1\right )^{7} a^{4}}\) | \(39\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {86 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5 a}+\frac {424 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{35 a}-\frac {86 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{5 a}+\frac {4 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}-\frac {2 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{7}}\) | \(91\) |
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {{\left (16 \, \cos \left (x\right )^{6} + 8 \, \cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} + 5\right )} \sin \left (x\right )}{35 \, a^{4} \cos \left (x\right )^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (36) = 72\).
Time = 5.40 (sec) , antiderivative size = 675, normalized size of antiderivative = 18.24 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=- \frac {70 \tan ^{13}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {140 \tan ^{11}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {602 \tan ^{9}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {424 \tan ^{7}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {602 \tan ^{5}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {140 \tan ^{3}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {70 \tan {\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} \]
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Time = 0.50 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {5 \, \tan \left (x\right )^{7} + 21 \, \tan \left (x\right )^{5} + 35 \, \tan \left (x\right )^{3} + 35 \, \tan \left (x\right )}{35 \, a^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {5 \, \tan \left (x\right )^{7} + 21 \, \tan \left (x\right )^{5} + 35 \, \tan \left (x\right )^{3} + 35 \, \tan \left (x\right )}{35 \, a^{4}} \]
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Time = 12.85 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx=\frac {\mathrm {tan}\left (x\right )}{a^4}+\frac {{\mathrm {tan}\left (x\right )}^3}{a^4}+\frac {3\,{\mathrm {tan}\left (x\right )}^5}{5\,a^4}+\frac {{\mathrm {tan}\left (x\right )}^7}{7\,a^4} \]
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