Numerically computing an oscillatory integral

Question

I want to numerically compute integrals of the form $$F(s) = \frac{1}{2\pi} \int_0^A \frac{e^{isu}}{1 + B\cos(r u)}\,du$$ for fixed $A,r>0, 1 > B >0$ for a finite number of values of $s\in \mathbb{R}$. I am familiar with the Filon method to deal with the highly oscillating case ($|s|\gg 1$). However, I am also concerned with the evaluations with slow oscillations ($0 < |s| \ll 1$), and I'm not sure if Filon method will be enough as it failed in several instances. Is it enough to do a naive approach of applying Filon method for large frequencies, otherwise use the default integral command?

Answer 1

You might write this as a series in powers of $B$, which converges since $0 < B < 1$:

$$ F(s) = \frac{1}{2\pi}\sum_{k=0}^\infty (-B)^k \int_0^A e^{isu} \cos(ru)^k\; du $$ where each term can be computed in closed form.