at.algebraic topology - How To Calculate A Winding Number?

We have a closed curve C on the plane given by parametric equations: x=x(t), y=y(t), t changes between a and b, x and y are smooth functions.
We want to calculate the winding number of this curve around the origin.
The most natural way to do it is to calculate the path integral:

$$int_C frac{-y,dx+x,dy}{x^2+y^2}$$

However, this integral turns out to be too complicated to calculate. What should we do now? Are there any efficient and strong methods to quickly and calculate the winding number?

Thanks.