$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$$
Does a closed form for the above exist, ideally for $n,m\in\mathbb{C}$ (most bounds probably removed at one point using analystic continuation)? The answer will probably have singularities whenever $\exists x \in\mathbb{R}$ such that $x^m=-1$, i.e. $m$ is an odd integer.
Closed forms exist for similar integrals, which is why I had hope in the answer existing. I came across this when trying to construct the Fourier series for $f(x)=\frac{1}{1+\tan^m(x)}$.
