Integrand size = 11, antiderivative size = 8 \[ \int \frac {2}{(\cos (x)-\sin (x))^2} \, dx=\frac {2}{-1+\cot (x)} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.62, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 3154} \[ \int \frac {2}{(\cos (x)-\sin (x))^2} \, dx=\frac {2 \sin (x)}{\cos (x)-\sin (x)} \]
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Rule 12
Rule 3154
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {1}{(\cos (x)-\sin (x))^2} \, dx \\ & = \frac {2 \sin (x)}{\cos (x)-\sin (x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.62 \[ \int \frac {2}{(\cos (x)-\sin (x))^2} \, dx=\frac {2 \sin (x)}{\cos (x)-\sin (x)} \]
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Time = 0.14 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12
method | result | size |
default | \(-\frac {2}{\tan \left (x \right )-1}\) | \(9\) |
risch | \(\frac {2}{{\mathrm e}^{2 i x}-i}\) | \(13\) |
parallelrisch | \(\frac {2 \sin \left (x \right )}{\cos \left (x \right )-\sin \left (x \right )}\) | \(14\) |
norman | \(-\frac {4 \tan \left (\frac {x}{2}\right )}{\tan \left (\frac {x}{2}\right )^{2}+2 \tan \left (\frac {x}{2}\right )-1}\) | \(23\) |
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none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.88 \[ \int \frac {2}{(\cos (x)-\sin (x))^2} \, dx=\frac {\cos \left (x\right ) + \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (5) = 10\).
Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 2.75 \[ \int \frac {2}{(\cos (x)-\sin (x))^2} \, dx=- \frac {4 \tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 2 \tan {\left (\frac {x}{2} \right )} - 1} \]
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none
Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {2}{(\cos (x)-\sin (x))^2} \, dx=-\frac {2}{\tan \left (x\right ) - 1} \]
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none
Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {2}{(\cos (x)-\sin (x))^2} \, dx=-\frac {2}{\tan \left (x\right ) - 1} \]
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Time = 15.12 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.62 \[ \int \frac {2}{(\cos (x)-\sin (x))^2} \, dx=\frac {2\,\sin \left (x\right )}{\cos \left (x\right )-\sin \left (x\right )} \]
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