Woww! Integratable

Calculus Level 3

\[ \large \int_0^{2\pi} \dfrac1{(1+2^{\sin x})(1+2^{\cos x})} \, dx \]

If the integral above equals to \( \dfrac {\pi^A}{B} \) for positive integers \(A\) and \(B\), find the value of \(A+B\).

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