Integrand size = 10, antiderivative size = 12 \[ \int \frac {\cot (x)}{\cot (x)+\csc (x)} \, dx=x-\frac {\sin (x)}{1+\cos (x)} \]
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Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4477, 2814, 2727} \[ \int \frac {\cot (x)}{\cot (x)+\csc (x)} \, dx=x-\frac {\sin (x)}{\cos (x)+1} \]
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Rule 2727
Rule 2814
Rule 4477
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (x)}{1+\cos (x)} \, dx \\ & = x-\int \frac {1}{1+\cos (x)} \, dx \\ & = x-\frac {\sin (x)}{1+\cos (x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {\cot (x)}{\cot (x)+\csc (x)} \, dx=x-\tan \left (\frac {x}{2}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25
method | result | size |
default | \(-\tan \left (\frac {x}{2}\right )+2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )\) | \(15\) |
risch | \(x -\frac {2 i}{{\mathrm e}^{i x}+1}\) | \(15\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.42 \[ \int \frac {\cot (x)}{\cot (x)+\csc (x)} \, dx=\frac {x \cos \left (x\right ) + x - \sin \left (x\right )}{\cos \left (x\right ) + 1} \]
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\[ \int \frac {\cot (x)}{\cot (x)+\csc (x)} \, dx=\int \frac {\cot {\left (x \right )}}{\cot {\left (x \right )} + \csc {\left (x \right )}}\, dx \]
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none
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.92 \[ \int \frac {\cot (x)}{\cot (x)+\csc (x)} \, dx=-\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {\cot (x)}{\cot (x)+\csc (x)} \, dx=x - \tan \left (\frac {1}{2} \, x\right ) \]
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Time = 23.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {\cot (x)}{\cot (x)+\csc (x)} \, dx=x-\mathrm {tan}\left (\frac {x}{2}\right ) \]
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