Integrand size = 11, antiderivative size = 12 \[ \int \frac {\csc (x)}{a+a \csc (x)} \, dx=-\frac {\cot (x)}{a+a \csc (x)} \]
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Time = 0.02 (sec) , antiderivative size = 12, 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 = {3879} \[ \int \frac {\csc (x)}{a+a \csc (x)} \, dx=-\frac {\cot (x)}{a \csc (x)+a} \]
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Rule 3879
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (x)}{a+a \csc (x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(26\) vs. \(2(12)=24\).
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.17 \[ \int \frac {\csc (x)}{a+a \csc (x)} \, dx=\frac {2 \sin \left (\frac {x}{2}\right )}{a \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )} \]
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Time = 0.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17
method | result | size |
default | \(-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(14\) |
norman | \(-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(14\) |
parallelrisch | \(-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(14\) |
risch | \(-\frac {2}{\left (i+{\mathrm e}^{i x}\right ) a}\) | \(16\) |
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none
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \frac {\csc (x)}{a+a \csc (x)} \, dx=-\frac {\cos \left (x\right ) - \sin \left (x\right ) + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \]
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\[ \int \frac {\csc (x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\csc {\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \]
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none
Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \frac {\csc (x)}{a+a \csc (x)} \, dx=-\frac {2}{a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1}} \]
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {\csc (x)}{a+a \csc (x)} \, dx=-\frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]
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Time = 17.55 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {\csc (x)}{a+a \csc (x)} \, dx=-\frac {2}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]
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