I was trying to solve an elementary indefinite integral but got stuck after a few steps. I want to know how to simplify the terms of the integration problem so that it can be evaluated in an indefinite form.
$$\int\log\tan(\frac{\pi}{4}+\frac{x}{2})dx$$.
Here is how I tried to simplify the problem:-
$$\int\log\tan(\frac{\pi}{4}+\frac{x}{2})\ dx\\=2\int\frac{\log(z)}{1+z^2}\ dz \space \space\text { , for } \mathrm{(\frac{\pi}{4}+\frac{x}{2}) = tan^{-1}(z)}\\ =2\int\frac{ye^y}{1+e^{2y}}\ dy \space \space\text { , for } \mathrm{z = e^y}\\ = 2y\tan^{-1}(e^y)-2\int\tan^{-1}(e^y) \ dy \ \ \ \text{ ,using by parts method of integration} \\= 2y\tan^{-1}(e^y)-2v\log|\tan{v}|+4\int\log |\tan{v}|\ dv \space \space\text { , for } \mathrm{y = \log(\tan{v})} $$
If I go with substitution method the expression will be clumsy and also I am not sure if I am missing something in this problem. I need to evaluate this indefinite integral but is there any technique to do this problem in short-cut method? Also alternative solutions are highly appreciated. Any help is valuable.