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

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 \(A+B+C+D\).

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