Integrand size = 6, antiderivative size = 9 \[ \int \log (\cos (x)) \tan (x) \, dx=-\frac {1}{2} \log ^2(\cos (x)) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3556, 4424, 2338} \[ \int \log (\cos (x)) \tan (x) \, dx=-\frac {1}{2} \log ^2(\cos (x)) \]
[In]
[Out]
Rule 2338
Rule 3556
Rule 4424
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\cos (x)\right ) \\ & = -\frac {1}{2} \log ^2(\cos (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \log (\cos (x)) \tan (x) \, dx=-\frac {1}{2} \log ^2(\cos (x)) \]
[In]
[Out]
Time = 1.13 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {\ln \left (\cos \left (x \right )\right )^{2}}{2}\) | \(8\) |
default | \(-\frac {\ln \left (\cos \left (x \right )\right )^{2}}{2}\) | \(8\) |
risch | \(-i x \ln \left (2\right )+\frac {x \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )}{2}-\frac {x \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}}{2}-\frac {x \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}}{2}+\frac {x \pi \operatorname {csgn}\left (i \cos \left (x \right )\right )^{3}}{2}+\frac {i \pi \ln \left (1+{\mathrm e}^{2 i x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )}{2}-\frac {i \pi \ln \left (1+{\mathrm e}^{2 i x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}}{2}-\frac {i \pi \ln \left (1+{\mathrm e}^{2 i x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}}{2}-i \left (i \ln \left (1+{\mathrm e}^{2 i x}\right )+x \right ) \ln \left ({\mathrm e}^{i x}\right )+\frac {i \pi \ln \left (1+{\mathrm e}^{2 i x}\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{3}}{2}+\ln \left (2\right ) \ln \left (1+{\mathrm e}^{2 i x}\right )-\frac {x^{2}}{2}-\frac {\ln \left (1+{\mathrm e}^{2 i x}\right )^{2}}{2}\) | \(262\) |
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \log (\cos (x)) \tan (x) \, dx=-\frac {1}{2} \, \log \left (\cos \left (x\right )\right )^{2} \]
[In]
[Out]
Time = 1.62 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \log (\cos (x)) \tan (x) \, dx=- \frac {\log {\left (\cos {\left (x \right )} \right )}^{2}}{2} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \log (\cos (x)) \tan (x) \, dx=-\frac {1}{2} \, \log \left (\cos \left (x\right )\right )^{2} \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \log (\cos (x)) \tan (x) \, dx=-\frac {1}{2} \, \log \left (\cos \left (x\right )\right )^{2} \]
[In]
[Out]
Time = 1.68 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \log (\cos (x)) \tan (x) \, dx=-\frac {{\ln \left (\cos \left (x\right )\right )}^2}{2} \]
[In]
[Out]