calculus - Tangent half-angle substitution for $int_0^{2 pi} frac{1- cos x}{3 + cos x}$

I want to evaluate the following integral using the tangent half-angle substitution $t = \tan (\frac{x}{2})$:

$$\int_0^{2 \pi} \frac{1- \cos x}{3 + \cos x} ~dx$$ However, making the substitution gives me $0$ for each of the limits of integration, which is obviously incorrect. I know that one way to solve this problem is to notice that, by symmetry, the equivalent integral 2$\int_0^{\pi} \frac{1- \cos x}{3 + \cos x} ~dx$ allows the subsitution to work. What are ways to make this substitution work without noticing this symmetry? I know that this general question has been asked on here before, but I am specifically interested in how I can make it work with this substitution.