Integrand size = 16, antiderivative size = 7 \[ \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\text {arctanh}(\sin (x))}{a^2} \]
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Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 3855} \[ \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\text {arctanh}(\sin (x))}{a^2} \]
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Rule 3254
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec (x) \, dx}{a^2} \\ & = \frac {\text {arctanh}(\sin (x))}{a^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\text {arctanh}(\sin (x))}{a^2} \]
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Time = 0.32 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\sin \left (x \right )\right )}{a^{2}}\) | \(8\) |
parallelrisch | \(\frac {-\ln \left (\tan \left (\frac {x}{2}\right )-1\right )+\ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{a^{2}}\) | \(22\) |
norman | \(\frac {\ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{a^{2}}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{a^{2}}\) | \(25\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{a^{2}}-\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{a^{2}}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (7) = 14\).
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.86 \[ \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\log \left (\sin \left (x\right ) + 1\right ) - \log \left (-\sin \left (x\right ) + 1\right )}{2 \, a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (7) = 14\).
Time = 2.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 3.14 \[ \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=- \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{a^{2}} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (7) = 14\).
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 3.00 \[ \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\log \left (\sin \left (x\right ) + 1\right )}{2 \, a^{2}} - \frac {\log \left (\sin \left (x\right ) - 1\right )}{2 \, a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (7) = 14\).
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 3.29 \[ \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\log \left (\sin \left (x\right ) + 1\right )}{2 \, a^{2}} - \frac {\log \left (-\sin \left (x\right ) + 1\right )}{2 \, a^{2}} \]
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Time = 0.06 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx=\frac {\mathrm {atanh}\left (\sin \left (x\right )\right )}{a^2} \]
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