Evaluate this definite integral
$$\int_0^{\frac{\pi}{2}}\frac{\cos(x)}{-\ln(\tan\left(\frac{x}{2}\right))\cos^4\left(\frac{x}{2}\right)}dx$$
I used Weierstrass substitution and then Feynman's integration technique (Differentiation under the integral sign) and then got the answer $2\ln(3)$ and I'm interested to see if there are any faster or alternate methods to evaluate this integral.
Edit: I wrote $\tan\frac{\pi}{2}$ originally instead of $\tan\frac{x}{2}$ and it's now corrected
Best Answer
Substitute $t= -\ln(\tan\frac{x}{2})$ to replace the inconvenient log function in the denominator
\begin{align} I= \int_0^{\frac{\pi}{2}}\frac{\cos x}{-\ln(\tan\frac{x}{2})\cos^4\frac{x}{2}}dx =2\int_0^\infty \frac{e^{-t} -e^{-3t}}t dt \end{align}
Then, integrate by parts
\begin{align} I &= 2\ln t(e^{-t} -e^{-3t})|_0^\infty -2\int_0^\infty\ln t(-e^{-t} +3e^{-3t})dt \\ & \overset {u=3t} = 2\int_0^\infty\ln t \> e^{-t}dt - 2\int_0^\infty (\ln u -\ln3) \>e^{-u}du \\ &= 2\ln3 \int_0^\infty e^{-u}du= 2\ln3 \end{align}