Integrand size = 8, antiderivative size = 6 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log (\log (\cos (x))) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4424, 2339, 29} \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log (\log (\cos (x))) \]
[In]
[Out]
Rule 29
Rule 2339
Rule 4424
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (\cos (x))\right ) \\ & = -\log (\log (\cos (x))) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log (\log (\cos (x))) \]
[In]
[Out]
Time = 0.92 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(-\ln \left (\ln \left (\cos \left (x \right )\right )\right )\) | \(7\) |
default | \(-\ln \left (\ln \left (\cos \left (x \right )\right )\right )\) | \(7\) |
risch | \(-\ln \left (\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) \operatorname {csgn}\left (i \cos \left (x \right )\right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (i \cos \left (x \right )\right )^{3}}{2}+\ln \left (2\right )-\ln \left (1+{\mathrm e}^{2 i x}\right )+\ln \left ({\mathrm e}^{i x}\right )\right )\) | \(109\) |
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log \left (\log \left (\cos \left (x\right )\right )\right ) \]
[In]
[Out]
\[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=\int \frac {\tan {\left (x \right )}}{\log {\left (\cos {\left (x \right )} \right )}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log \left (\log \left (\cos \left (x\right )\right )\right ) \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\log \left ({\left | \log \left (\cos \left (x\right )\right ) \right |}\right ) \]
[In]
[Out]
Time = 1.59 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\log (\cos (x))} \, dx=-\ln \left (\ln \left (\cos \left (x\right )\right )\right ) \]
[In]
[Out]