This one is familiar to?

$\large\int\limits_{0}^{\pi/4096} \ln \left(\frac{\tan (2048x)-\tan (1024 x)}{\tan(1024 x)}\right)\,\mathrm dx$

The value of above integral is $\large{\frac{\pi^{A}}{B}(\ln (C))^{D}}$ for some positive integers $A,B,C,D$ .

Find the minimum value of $A+B+C+D$ .