Integrand size = 2, antiderivative size = 3 \[ \int \sec (x) \, dx=\text {arctanh}(\sin (x)) \]
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Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3855} \[ \int \sec (x) \, dx=\text {arctanh}(\sin (x)) \]
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Rule 3855
Rubi steps \begin{align*} \text {integral}& = \text {arctanh}(\sin (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \sec (x) \, dx=\text {arctanh}(\sin (x)) \]
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Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 2.33
method | result | size |
default | \(\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )\) | \(7\) |
norman | \(-\ln \left (\tan \left (\frac {x}{2}\right )-1\right )+\ln \left (1+\tan \left (\frac {x}{2}\right )\right )\) | \(18\) |
parallelrisch | \(-\ln \left (\tan \left (\frac {x}{2}\right )-1\right )+\ln \left (1+\tan \left (\frac {x}{2}\right )\right )\) | \(18\) |
risch | \(\ln \left (i+{\mathrm e}^{i x}\right )-\ln \left ({\mathrm e}^{i x}-i\right )\) | \(22\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (3) = 6\).
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 5.67 \[ \int \sec (x) \, dx=\frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (3) = 6\).
Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 5.00 \[ \int \sec (x) \, dx=- \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (3) = 6\).
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 5.00 \[ \int \sec (x) \, dx=\frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{2} \, \log \left (\sin \left (x\right ) - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (3) = 6\).
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 5.67 \[ \int \sec (x) \, dx=\frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Time = 0.05 (sec) , antiderivative size = 11, normalized size of antiderivative = 3.67 \[ \int \sec (x) \, dx=\ln \left (\frac {1}{\cos \left (x\right )}\right )+\ln \left (\sin \left (x\right )+1\right ) \]
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