Vector Calculus · Superchallenge problem

Let $\mathbf F = \left(y - \dfrac{y}{x^2 + y^2},\; x + \dfrac{x}{x^2 + y^2}\right)$ and let $C$ be the spiral $\mathbf r(t) = (t\cos t,\; t\sin t)$ for $\dfrac{\pi}{2}\le t\le 4\pi$. Evaluate $\displaystyle\int_C\mathbf F\cdot d\mathbf r$. Give your answer in terms of $\pi$.