Integrand size = 10, antiderivative size = 9 \[ \int \frac {\cot (x)}{\sec (x)+\tan (x)} \, dx=-x-\text {arctanh}(\cos (x)) \]
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Time = 0.09 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4476, 2918, 3855, 8} \[ \int \frac {\cot (x)}{\sec (x)+\tan (x)} \, dx=-\text {arctanh}(\cos (x))-x \]
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Rule 8
Rule 2918
Rule 3855
Rule 4476
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (x) \cot (x)}{1+\sin (x)} \, dx \\ & = -\int 1 \, dx+\int \csc (x) \, dx \\ & = -x-\text {arctanh}(\cos (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(20\) vs. \(2(9)=18\).
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.22 \[ \int \frac {\cot (x)}{\sec (x)+\tan (x)} \, dx=-x-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right ) \]
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Time = 0.36 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.56
method | result | size |
default | \(\ln \left (\tan \left (\frac {x}{2}\right )\right )-2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )\) | \(14\) |
risch | \(-x +\ln \left ({\mathrm e}^{i x}-1\right )-\ln \left ({\mathrm e}^{i x}+1\right )\) | \(23\) |
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (9) = 18\).
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 2.44 \[ \int \frac {\cot (x)}{\sec (x)+\tan (x)} \, dx=-x - \frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \]
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\[ \int \frac {\cot (x)}{\sec (x)+\tan (x)} \, dx=\int \frac {\cot {\left (x \right )}}{\tan {\left (x \right )} + \sec {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (9) = 18\).
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.56 \[ \int \frac {\cot (x)}{\sec (x)+\tan (x)} \, dx=-2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) + \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {\cot (x)}{\sec (x)+\tan (x)} \, dx=-x + \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]
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Time = 22.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.56 \[ \int \frac {\cot (x)}{\sec (x)+\tan (x)} \, dx=2\,\mathrm {atan}\left (\frac {8}{4\,\mathrm {tan}\left (\frac {x}{2}\right )+4}-1\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right ) \]
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