\[ \large\int_0^1 \frac{\sqrt{1-x^2}}{x^8+1}\,dx \; = \; \tfrac{\pi^A}{\sqrt{B}}\left [ \sqrt{\cos\left(\tfrac{\pi}{C}\right)} \cos\left(\tfrac{D\pi}{E}\right) - \sqrt{\sin\left(\tfrac{\pi}{C}\right)} \sin\left(\tfrac{F\pi}{E}\right)\right ] \]

Given that the equation above holds true, where \(A,B,C,D,E\) and \(F\) are positive integers such that \(\tfrac{F\pi}{E} < \tfrac{\pi}{C} < \tfrac{D\pi}{E}\) are acute angles, and \(D,E,F\) are pairwise coprime, find the value of \(A+B+C+D+E+F\).

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