Integrand size = 8, antiderivative size = 9 \[ \int \csc (x) \log (\tan (x)) \sec (x) \, dx=\frac {1}{2} \log ^2(\tan (x)) \]
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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 = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2700, 29, 6818} \[ \int \csc (x) \log (\tan (x)) \sec (x) \, dx=\frac {1}{2} \log ^2(\tan (x)) \]
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Rule 29
Rule 2700
Rule 6818
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \log ^2(\tan (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \csc (x) \log (\tan (x)) \sec (x) \, dx=\frac {1}{2} \log ^2(\tan (x)) \]
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Time = 1.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\ln \left (\tan \left (x \right )\right )^{2}}{2}\) | \(8\) |
default | \(\frac {\ln \left (\tan \left (x \right )\right )^{2}}{2}\) | \(8\) |
risch | \(\text {Expression too large to display}\) | \(764\) |
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none
Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.33 \[ \int \csc (x) \log (\tan (x)) \sec (x) \, dx=\frac {1}{2} \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{2} \]
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\[ \int \csc (x) \log (\tan (x)) \sec (x) \, dx=\int \log {\left (\tan {\left (x \right )} \right )} \csc {\left (x \right )} \sec {\left (x \right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \csc (x) \log (\tan (x)) \sec (x) \, dx=\frac {1}{2} \, \log \left (\tan \left (x\right )\right )^{2} \]
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none
Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \csc (x) \log (\tan (x)) \sec (x) \, dx=\frac {1}{2} \, \log \left (\tan \left (x\right )\right )^{2} \]
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Time = 29.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 3.00 \[ \int \csc (x) \log (\tan (x)) \sec (x) \, dx=\frac {{\ln \left (-\frac {{\mathrm {e}}^{x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}+1}\right )}^2}{2} \]
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