How to find the primitives of $\frac{1}{p(\sin x)q(\cos x)}$

Question

Is it always possible to explicitly evaluate $$\int\frac{1}{p(\sin{x})q(\cos{x})}dx$$ with $p(x),q(x)\in\mathbb{Q}[x] $?

Answer 1

By letting $x=2\arctan\frac{t}{2}$ your problem boils down to the integration of a rational function.
By partial fraction decomposition, such problem can be solved by locating the roots of a polynomial and their multiplicity.