I want to find the winding number of the closed curve $\gamma(t)=\frac{-1}{2}+\cos(3t)+i\sin(4t)$ at point $0$ ; where $0\leq t\leq 2\pi$. For that, we have to find the value of
$$ \int_{0}^{2\pi} \frac{-3\sin(3t)+4i\cos(4t)}{\frac{-1}{2}+\cos(3t)+i\sin(4t)}dt. $$
I have no idea how to find the above integral. Also the given closed curve is too complex to visualise so we can't find the the winding number geometrically at $0$. Is there any simpler method to find the integral or winding number of the above curve at $0$ ? Thanks in advance.
What about drawing and counting cutting numbers, see Crazy calculation for winding numbers?
Hence the winding number is $-1$.