Integrand size = 8, antiderivative size = 9 \[ \int \csc (2 x) \log (\tan (x)) \, dx=\frac {1}{4} \log ^2(\tan (x)) \]
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Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3855, 6818} \[ \int \csc (2 x) \log (\tan (x)) \, dx=\frac {1}{4} \log ^2(\tan (x)) \]
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Rule 3855
Rule 6818
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \log ^2(\tan (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \csc (2 x) \log (\tan (x)) \, dx=\frac {1}{4} \log ^2(\tan (x)) \]
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Time = 0.60 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\ln \left (\tan \left (x \right )\right )^{2}}{4}\) | \(8\) |
default | \(\frac {\ln \left (\tan \left (x \right )\right )^{2}}{4}\) | \(8\) |
norman | \(\frac {\ln \left (\tan \left (x \right )\right )^{2}}{4}\) | \(8\) |
parallelrisch | \(\frac {\ln \left (\tan \left (x \right )\right )^{2}}{4}\) | \(8\) |
risch | \(\text {Expression too large to display}\) | \(764\) |
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Time = 0.23 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \csc (2 x) \log (\tan (x)) \, dx=\frac {1}{4} \, \log \left (\tan \left (x\right )\right )^{2} \]
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Timed out. \[ \int \csc (2 x) \log (\tan (x)) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (7) = 14\).
Time = 0.31 (sec) , antiderivative size = 265, normalized size of antiderivative = 29.44 \[ \int \csc (2 x) \log (\tan (x)) \, dx=\frac {1}{4} \, {\left (\pi - 2 \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 2 \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + \frac {1}{4} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right )^{2} - \frac {1}{4} \, {\left (\pi - 2 \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + \frac {1}{4} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )^{2} - \frac {1}{4} \, \pi \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + \frac {1}{4} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )^{2} + \frac {1}{8} \, {\left (\log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) - \frac {1}{16} \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )^{2} - \frac {1}{16} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )^{2} - \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - \frac {1}{16} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )^{2} - \frac {1}{2} \, \log \left (\cot \left (2 \, x\right ) + \csc \left (2 \, x\right )\right ) \log \left (\tan \left (x\right )\right ) \]
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\[ \int \csc (2 x) \log (\tan (x)) \, dx=\int { \csc \left (2 \, x\right ) \log \left (\tan \left (x\right )\right ) \,d x } \]
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Time = 27.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 3.00 \[ \int \csc (2 x) \log (\tan (x)) \, dx=\frac {{\ln \left (-\frac {{\mathrm {e}}^{x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}+1}\right )}^2}{4} \]
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