Integrand size = 11, antiderivative size = 16 \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx=-x-\frac {\sin (x)}{1-\cos (x)} \]
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Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2814, 2727} \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx=-x-\frac {\sin (x)}{1-\cos (x)} \]
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Rule 2727
Rule 2814
Rubi steps \begin{align*} \text {integral}& = -x+\int \frac {1}{1-\cos (x)} \, dx \\ & = -x-\frac {\sin (x)}{1-\cos (x)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx=-\csc (x) \left (1+\cos (x)+2 \arcsin \left (\sqrt {\sin ^2\left (\frac {x}{2}\right )}\right ) \sqrt {\sin ^2(x)}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
default | \(-2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {1}{\tan \left (\frac {x}{2}\right )}\) | \(17\) |
risch | \(-x -\frac {2 i}{{\mathrm e}^{i x}-1}\) | \(17\) |
parallelrisch | \(\frac {-x \tan \left (\frac {x}{2}\right )-1}{\tan \left (\frac {x}{2}\right )}\) | \(17\) |
norman | \(\frac {-1-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )-x \tan \left (\frac {x}{2}\right )-\left (\tan ^{3}\left (\frac {x}{2}\right )\right ) x}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )}\) | \(44\) |
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx=-\frac {x \sin \left (x\right ) + \cos \left (x\right ) + 1}{\sin \left (x\right )} \]
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Time = 0.21 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.50 \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx=- x - \frac {1}{\tan {\left (\frac {x}{2} \right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx=-\frac {\cos \left (x\right ) + 1}{\sin \left (x\right )} - 2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx=-x - \frac {1}{\tan \left (\frac {1}{2} \, x\right )} \]
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Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {\cos (x)}{1-\cos (x)} \, dx=-x-\mathrm {cot}\left (\frac {x}{2}\right ) \]
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