Integrand size = 16, antiderivative size = 33 \[ \int \frac {\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3 x}{8 a^2}+\frac {3 \cos (x) \sin (x)}{8 a^2}+\frac {\cos ^3(x) \sin (x)}{4 a^2} \]
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Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3254, 2715, 8} \[ \int \frac {\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3 x}{8 a^2}+\frac {\sin (x) \cos ^3(x)}{4 a^2}+\frac {3 \sin (x) \cos (x)}{8 a^2} \]
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Rule 8
Rule 2715
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^4(x) \, dx}{a^2} \\ & = \frac {\cos ^3(x) \sin (x)}{4 a^2}+\frac {3 \int \cos ^2(x) \, dx}{4 a^2} \\ & = \frac {3 \cos (x) \sin (x)}{8 a^2}+\frac {\cos ^3(x) \sin (x)}{4 a^2}+\frac {3 \int 1 \, dx}{8 a^2} \\ & = \frac {3 x}{8 a^2}+\frac {3 \cos (x) \sin (x)}{8 a^2}+\frac {\cos ^3(x) \sin (x)}{4 a^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\frac {3 x}{8}+\frac {1}{4} \sin (2 x)+\frac {1}{32} \sin (4 x)}{a^2} \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {12 x +\sin \left (4 x \right )+8 \sin \left (2 x \right )}{32 a^{2}}\) | \(20\) |
risch | \(\frac {3 x}{8 a^{2}}+\frac {\sin \left (4 x \right )}{32 a^{2}}+\frac {\sin \left (2 x \right )}{4 a^{2}}\) | \(26\) |
default | \(\frac {\frac {\tan \left (x \right )}{4 \left (1+\tan ^{2}\left (x \right )\right )^{2}}+\frac {3 \tan \left (x \right )}{8 \left (1+\tan ^{2}\left (x \right )\right )}+\frac {3 \arctan \left (\tan \left (x \right )\right )}{8}}{a^{2}}\) | \(35\) |
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {{\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + 3 \, x}{8 \, a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (34) = 68\).
Time = 20.78 (sec) , antiderivative size = 549, normalized size of antiderivative = 16.64 \[ \int \frac {\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3 x \tan ^{8}{\left (\frac {x}{2} \right )}}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} + \frac {12 x \tan ^{6}{\left (\frac {x}{2} \right )}}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} + \frac {18 x \tan ^{4}{\left (\frac {x}{2} \right )}}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} + \frac {12 x \tan ^{2}{\left (\frac {x}{2} \right )}}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} + \frac {3 x}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} - \frac {10 \tan ^{7}{\left (\frac {x}{2} \right )}}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} + \frac {6 \tan ^{5}{\left (\frac {x}{2} \right )}}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} - \frac {6 \tan ^{3}{\left (\frac {x}{2} \right )}}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} + \frac {10 \tan {\left (\frac {x}{2} \right )}}{8 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 48 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 32 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 8 a^{2}} \]
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Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3 \, \tan \left (x\right )^{3} + 5 \, \tan \left (x\right )}{8 \, {\left (a^{2} \tan \left (x\right )^{4} + 2 \, a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} + \frac {3 \, x}{8 \, a^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3 \, x}{8 \, a^{2}} + \frac {3 \, \tan \left (x\right )^{3} + 5 \, \tan \left (x\right )}{8 \, {\left (\tan \left (x\right )^{2} + 1\right )}^{2} a^{2}} \]
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Time = 13.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^8(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3\,x}{8\,a^2}+\frac {3\,\cos \left (x\right )\,{\sin \left (x\right )}^3}{8\,a^2}+\frac {5\,{\cos \left (x\right )}^3\,\sin \left (x\right )}{8\,a^2} \]
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