Vector Calculus · Superchallenge problem

For a point $(a, 0)$ let $\mathbf V_a = \left(\dfrac{-y}{(x - a)^2 + y^2},\; \dfrac{x - a}{(x - a)^2 + y^2}\right)$, and let $\mathbf F = \mathbf V_1 - \mathbf V_{-1} + (0,\; x^2)$. Let $C$ be the lemniscate $(x^2 + y^2)^2 = 2(x^2 - y^2)$, traced once by $\mathbf r(t) = \left(\dfrac{\sqrt2\cos t}{1 + \sin^2 t},\; \dfrac{\sqrt2\sin t\cos t}{1 + \sin^2 t}\right)$, $0\le t\le 2\pi$. Evaluate $\displaystyle\oint_C\mathbf F\cdot d\mathbf r$. Give your answer in terms of $\pi$.