Integrand size = 11, antiderivative size = 14 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {i \csc (x)}{i+\cot (x)} \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3569} \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {i \csc (x)}{\cot (x)+i} \]
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Rule 3569
Rubi steps \begin{align*} \text {integral}& = \frac {i \csc (x)}{i+\cot (x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=i \cos (x)+\sin (x) \]
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Time = 0.10 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64
method | result | size |
risch | \(i {\mathrm e}^{-i x}\) | \(9\) |
default | \(\frac {2}{-i+\tan \left (\frac {x}{2}\right )}\) | \(12\) |
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none
Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.43 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=i \, e^{\left (-i \, x\right )} \]
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Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {i \csc {\left (x \right )}}{\cot {\left (x \right )} + i} \]
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none
Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {2}{\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - i} \]
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {2}{\tan \left (\frac {1}{2} \, x\right ) - i} \]
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Time = 12.48 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {2{}\mathrm {i}}{1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}} \]
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