Integrand size = 8, antiderivative size = 12 \[ \int \log (\cos (x)) \sec ^2(x) \, dx=-x+\tan (x)+\log (\cos (x)) \tan (x) \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3852, 8, 2634, 3554} \[ \int \log (\cos (x)) \sec ^2(x) \, dx=-x+\tan (x)+\tan (x) \log (\cos (x)) \]
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Rule 8
Rule 2634
Rule 3554
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \log (\cos (x)) \tan (x)+\int \tan ^2(x) \, dx \\ & = \tan (x)+\log (\cos (x)) \tan (x)-\int 1 \, dx \\ & = -x+\tan (x)+\log (\cos (x)) \tan (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \log (\cos (x)) \sec ^2(x) \, dx=-x+\tan (x)+\log (\cos (x)) \tan (x) \]
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Time = 1.54 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
parallelrisch | \(-x +\tan \left (x \right )+\ln \left (\cos \left (x \right )\right ) \tan \left (x \right )\) | \(13\) |
norman | \(\frac {x -x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right ) \ln \left (\frac {1-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\tan ^{2}\left (\frac {x}{2}\right )}\right )-2 \tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}\) | \(57\) |
default | \(-4 i \left (\frac {\frac {{\mathrm e}^{2 i x} \ln \left (\left (1+{\mathrm e}^{2 i x}\right ) {\mathrm e}^{-i x}\right )}{2}-\frac {1}{2}}{1+{\mathrm e}^{2 i x}}-\frac {\ln \left (1+{\mathrm e}^{2 i x}\right )}{4}+\frac {\ln \left (2\right )}{2+2 \,{\mathrm e}^{2 i x}}\right )\) | \(67\) |
risch | \(-\frac {2 i \ln \left ({\mathrm e}^{i x}\right )}{1+{\mathrm e}^{2 i x}}+\frac {-i \ln \left (1+{\mathrm e}^{2 i x}\right ) {\mathrm e}^{2 i x}+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}+\pi \operatorname {csgn}\left (i \cos \left (x \right )\right )^{3}-2 x \,{\mathrm e}^{2 i x}-2 i \ln \left (2\right )+i \ln \left (1+{\mathrm e}^{2 i x}\right )+2 i-2 x}{1+{\mathrm e}^{2 i x}}\) | \(156\) |
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Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int \log (\cos (x)) \sec ^2(x) \, dx=-\frac {x \cos \left (x\right ) - \log \left (\cos \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right )} \]
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Time = 14.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \log (\cos (x)) \sec ^2(x) \, dx=- x + \log {\left (\cos {\left (x \right )} \right )} \tan {\left (x \right )} + \frac {\sin {\left (x \right )}}{\cos {\left (x \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (12) = 24\).
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 7.83 \[ \int \log (\cos (x)) \sec ^2(x) \, dx=-\frac {2 \, \log \left (-\frac {\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1}{\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}\right ) \sin \left (x\right )}{{\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}} - \frac {2 \, \sin \left (x\right )}{{\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}} - 2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \log (\cos (x)) \sec ^2(x) \, dx=\log \left (\cos \left (x\right )\right ) \tan \left (x\right ) - x + \tan \left (x\right ) \]
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Time = 1.87 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.92 \[ \int \log (\cos (x)) \sec ^2(x) \, dx=\mathrm {tan}\left (x\right )-2\,x+\ln \left (\cos \left (x\right )\right )\,\mathrm {tan}\left (x\right )+\ln \left (\cos \left (x\right )\right )\,1{}\mathrm {i}-\ln \left (\cos \left (2\,x\right )+1+\sin \left (2\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
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