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$\large \int_0^\pi \ln( 25\sin^2 x + 16\cos^2 x) \, dx = C \pi \ln\left(\frac AB\right)$

where $A$ and $B$ are co-prime integers and $C$ is also an integer. $C$ is greater than $1$.

Then, find $A^2+B^2+C^2$.

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