A calculus problem by Jose Sacramento

Calculus Level 4

\[ \large \int_{-\pi}^\pi \dfrac{\ln(1 + \sin^2 x)} {\ln(1 + \sin^2 x) + \ln(1 + \cos^2 x)} \, dx \]

Find the value of the closed form (to 3 decimal places) of the integral above.

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