Integrand size = 10, antiderivative size = 9 \[ \int \frac {\csc (x)}{\cot (x)+\csc (x)} \, dx=\frac {\sin (x)}{1+\cos (x)} \]
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Time = 0.03 (sec) , antiderivative size = 9, 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 = {3245, 2727} \[ \int \frac {\csc (x)}{\cot (x)+\csc (x)} \, dx=\frac {\sin (x)}{\cos (x)+1} \]
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Rule 2727
Rule 3245
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{1+\cos (x)} \, dx \\ & = \frac {\sin (x)}{1+\cos (x)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int \frac {\csc (x)}{\cot (x)+\csc (x)} \, dx=\tan \left (\frac {x}{2}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56
method | result | size |
default | \(\tan \left (\frac {x}{2}\right )\) | \(5\) |
risch | \(\frac {2 i}{{\mathrm e}^{i x}+1}\) | \(13\) |
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none
Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (x)}{\cot (x)+\csc (x)} \, dx=\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} \]
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\[ \int \frac {\csc (x)}{\cot (x)+\csc (x)} \, dx=\int \frac {\csc {\left (x \right )}}{\cot {\left (x \right )} + \csc {\left (x \right )}}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (x)}{\cot (x)+\csc (x)} \, dx=\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} \]
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none
Time = 0.28 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.44 \[ \int \frac {\csc (x)}{\cot (x)+\csc (x)} \, dx=\tan \left (\frac {1}{2} \, x\right ) \]
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Time = 23.16 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.44 \[ \int \frac {\csc (x)}{\cot (x)+\csc (x)} \, dx=\mathrm {tan}\left (\frac {x}{2}\right ) \]
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