Integrand size = 10, antiderivative size = 11 \[ \int \frac {1}{1-\cos (x)+\sin (x)} \, dx=-\log \left (1+\cot \left (\frac {x}{2}\right )\right ) \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3200, 31} \[ \int \frac {1}{1-\cos (x)+\sin (x)} \, dx=-\log \left (\cot \left (\frac {x}{2}\right )+1\right ) \]
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Rule 31
Rule 3200
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\cot \left (\frac {x}{2}\right )\right ) \\ & = -\log \left (1+\cot \left (\frac {x}{2}\right )\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(24\) vs. \(2(11)=22\).
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.18 \[ \int \frac {1}{1-\cos (x)+\sin (x)} \, dx=\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \]
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Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.45
method | result | size |
default | \(\ln \left (\tan \left (\frac {x}{2}\right )\right )-\ln \left (1+\tan \left (\frac {x}{2}\right )\right )\) | \(16\) |
norman | \(\ln \left (\tan \left (\frac {x}{2}\right )\right )-\ln \left (1+\tan \left (\frac {x}{2}\right )\right )\) | \(16\) |
parallelrisch | \(\ln \left (\tan \left (\frac {x}{2}\right )\right )-\ln \left (1+\tan \left (\frac {x}{2}\right )\right )\) | \(16\) |
risch | \(\ln \left ({\mathrm e}^{i x}-1\right )-\ln \left (i+{\mathrm e}^{i x}\right )\) | \(21\) |
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none
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {1}{1-\cos (x)+\sin (x)} \, dx=\frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) \]
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Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \frac {1}{1-\cos (x)+\sin (x)} \, dx=- \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} + \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (9) = 18\).
Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.27 \[ \int \frac {1}{1-\cos (x)+\sin (x)} \, dx=-\log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) + \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {1}{1-\cos (x)+\sin (x)} \, dx=-\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) + \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1-\cos (x)+\sin (x)} \, dx=-2\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right ) \]
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