Looks like an elementary integral!

Calculus Level 5

$\large{\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx = \pi^{A} \ln(B) }$

If the above integral is true for positive integers $$A,B$$, then evaluate $$B^A$$ .

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