let $m, n \in \mathbb{Z}$ be fixed and let $ 0 < r \neq 1$, determine the winding number of the closed curve $\gamma(t) : = e^{imt} + re^{int}$ $\gamma: [0, 2\pi] \to \mathbb{C} \backslash 0$ around zero

so we know that
$w\_\gamma (0) = \frac{1}{2\pi i} \int\_{\gamma} \frac{1}{z-0} dz$
would I then be having

$$w\_\gamma (0) = \frac{1}{2\pi i} \int\_{0}^{2\pi} \frac{ime^{imt} + in re^{int}}{e^{imt} + re^{int}} dt$$
?

is my approach correct until now and how could I proceed?