Integrand size = 16, antiderivative size = 29 \[ \int \frac {\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin (x)}{a^2}-\frac {2 \sin ^3(x)}{3 a^2}+\frac {\sin ^5(x)}{5 a^2} \]
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Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2713} \[ \int \frac {\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin ^5(x)}{5 a^2}-\frac {2 \sin ^3(x)}{3 a^2}+\frac {\sin (x)}{a^2} \]
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Rule 2713
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^5(x) \, dx}{a^2} \\ & = -\frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )}{a^2} \\ & = \frac {\sin (x)}{a^2}-\frac {2 \sin ^3(x)}{3 a^2}+\frac {\sin ^5(x)}{5 a^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5}}{a^2} \]
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Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{5}\left (x \right )\right )}{5}-\frac {2 \left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right )}{a^{2}}\) | \(20\) |
default | \(\frac {\frac {\left (\sin ^{5}\left (x \right )\right )}{5}-\frac {2 \left (\sin ^{3}\left (x \right )\right )}{3}+\sin \left (x \right )}{a^{2}}\) | \(20\) |
parallelrisch | \(\frac {150 \sin \left (x \right )+3 \sin \left (5 x \right )+25 \sin \left (3 x \right )}{240 a^{2}}\) | \(23\) |
risch | \(\frac {5 \sin \left (x \right )}{8 a^{2}}+\frac {\sin \left (5 x \right )}{80 a^{2}}+\frac {5 \sin \left (3 x \right )}{48 a^{2}}\) | \(27\) |
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {{\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right )}{15 \, a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (27) = 54\).
Time = 32.28 (sec) , antiderivative size = 362, normalized size of antiderivative = 12.48 \[ \int \frac {\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {30 \tan ^{9}{\left (\frac {x}{2} \right )}}{15 a^{2} \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a^{2}} + \frac {40 \tan ^{7}{\left (\frac {x}{2} \right )}}{15 a^{2} \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a^{2}} + \frac {116 \tan ^{5}{\left (\frac {x}{2} \right )}}{15 a^{2} \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a^{2}} + \frac {40 \tan ^{3}{\left (\frac {x}{2} \right )}}{15 a^{2} \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a^{2}} + \frac {30 \tan {\left (\frac {x}{2} \right )}}{15 a^{2} \tan ^{10}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{8}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{6}{\left (\frac {x}{2} \right )} + 150 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 75 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 15 a^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a^{2}} \]
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Time = 13.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\frac {{\sin \left (x\right )}^5}{5}-\frac {2\,{\sin \left (x\right )}^3}{3}+\sin \left (x\right )}{a^2} \]
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