Evaluate: $\int_{0}^{\pi} \frac{\cos x \cos{(A \cos x)}}{B \cos x +C \sin x}\,dx$

Question

$$\int_{0}^{\pi} \frac{\cos x \cos{(A \cos x)}}{B \cos x+C \sin x}\,dx$$

Is it possible to calculate this? (A, B, C are assumed to be real numbers.)

I would be grateful if you could share some of the good integration skills, references, or any advice.