Closed form for $\int \frac{1}{\sin^nx +\cos^mx}dx$

Question

Is

$$\int \frac{1}{\sin^nx +\cos^mx}dx, \qquad m,n \in \mathbb Z$$

Always expressible as a combination of elementary functions?

Answer 1

As remarked in the comments by CodeLabMaster, such integral boils down to the integral of a rational function through the substitution $x=2\arctan\frac{t}{2}$. By partial fraction decomposition, the primitive is a combination of rational functions and logarithms (arising from $\int\frac{dt}{t-\alpha}=\log(t-\alpha)$) and it has a "nice" closed form as soon as we are able to locate the roots of $$ p_{n,m}(t)=(2t)^n(1+t^2)^m+(1-t^2)^m(1+t^2)^n. $$