Integrand size = 12, antiderivative size = 11 \[ \int \frac {\sec (x)}{\sec (x)-\tan (x)} \, dx=\frac {\cos (x)}{1-\sin (x)} \]
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Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3244, 2727} \[ \int \frac {\sec (x)}{\sec (x)-\tan (x)} \, dx=\frac {\cos (x)}{1-\sin (x)} \]
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Rule 2727
Rule 3244
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{1-\sin (x)} \, dx \\ & = \frac {\cos (x)}{1-\sin (x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(25\) vs. \(2(11)=22\).
Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.27 \[ \int \frac {\sec (x)}{\sec (x)-\tan (x)} \, dx=\frac {2 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )} \]
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Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\) | \(11\) |
risch | \(\frac {2}{{\mathrm e}^{i x}-i}\) | \(13\) |
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none
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {\sec (x)}{\sec (x)-\tan (x)} \, dx=\frac {\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \]
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\[ \int \frac {\sec (x)}{\sec (x)-\tan (x)} \, dx=\int \frac {\sec {\left (x \right )}}{- \tan {\left (x \right )} + \sec {\left (x \right )}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\sec (x)}{\sec (x)-\tan (x)} \, dx=-\frac {2}{\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \]
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none
Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {\sec (x)}{\sec (x)-\tan (x)} \, dx=-\frac {2}{\tan \left (\frac {1}{2} \, x\right ) - 1} \]
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Time = 22.35 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {\sec (x)}{\sec (x)-\tan (x)} \, dx=-\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )-1} \]
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