Integrand size = 10, antiderivative size = 4 \[ \int \frac {1}{1-\cos ^2(x)} \, dx=-\cot (x) \]
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Time = 0.02 (sec) , antiderivative size = 4, 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 = {3254, 3852, 8} \[ \int \frac {1}{1-\cos ^2(x)} \, dx=-\cot (x) \]
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Rule 8
Rule 3254
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \int \csc ^2(x) \, dx \\ & = -\text {Subst}(\int 1 \, dx,x,\cot (x)) \\ & = -\cot (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1-\cos ^2(x)} \, dx=-\cot (x) \]
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Time = 0.09 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25
method | result | size |
parallelrisch | \(-\cot \left (x \right )\) | \(5\) |
default | \(-\frac {1}{\tan \left (x \right )}\) | \(7\) |
risch | \(-\frac {2 i}{{\mathrm e}^{2 i x}-1}\) | \(13\) |
norman | \(\frac {-\frac {1}{2}+\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}}{\tan \left (\frac {x}{2}\right )}\) | \(18\) |
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none
Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 2.00 \[ \int \frac {1}{1-\cos ^2(x)} \, dx=-\frac {\cos \left (x\right )}{\sin \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (3) = 6\).
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 3.50 \[ \int \frac {1}{1-\cos ^2(x)} \, dx=\frac {\tan {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tan {\left (\frac {x}{2} \right )}} \]
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none
Time = 0.21 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.50 \[ \int \frac {1}{1-\cos ^2(x)} \, dx=-\frac {1}{\tan \left (x\right )} \]
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none
Time = 0.31 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.50 \[ \int \frac {1}{1-\cos ^2(x)} \, dx=-\frac {1}{\tan \left (x\right )} \]
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Time = 2.66 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1-\cos ^2(x)} \, dx=-\mathrm {cot}\left (x\right ) \]
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