Integrand size = 9, antiderivative size = 9 \[ \int \frac {1}{\sec (x)-\tan (x)} \, dx=-\log (1-\sin (x)) \]
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Time = 0.04 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3238, 2746, 31} \[ \int \frac {1}{\sec (x)-\tan (x)} \, dx=-\log (1-\sin (x)) \]
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Rule 31
Rule 2746
Rule 3238
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (x)}{1-\sin (x)} \, dx \\ & = -\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,-\sin (x)\right ) \\ & = -\log (1-\sin (x)) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sec (x)-\tan (x)} \, dx=-2 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \]
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Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\ln \left (\sin \left (x \right )-1\right )\) | \(8\) |
risch | \(i x -2 \ln \left ({\mathrm e}^{i x}-i\right )\) | \(17\) |
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none
Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sec (x)-\tan (x)} \, dx=-\log \left (-\sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (7) = 14\).
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.89 \[ \int \frac {1}{\sec (x)-\tan (x)} \, dx=- \log {\left (\tan {\left (x \right )} - \sec {\left (x \right )} \right )} + \frac {\log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (9) = 18\).
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 3.22 \[ \int \frac {1}{\sec (x)-\tan (x)} \, dx=-2 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (9) = 18\).
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\sec (x)-\tan (x)} \, dx=\log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) \]
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Time = 22.96 (sec) , antiderivative size = 19, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\sec (x)-\tan (x)} \, dx=\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right ) \]
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