$\newcommand{\F}{\mathbf{F}}$\newcommand{\R}{\mathbb{R}}$Consider the vector field $\F(x,y)=\big(1-x^2y, xy^2+\exp(y^2)\cdot\cos(y)\big)$ for $(x,y)\in\R^2$ and the curves $C_1=\{(x,y)\in\R^2\colon x\in[-1,1], y=0\}\text{ and }C_2=\{(x,y)\in\R^2\colon x^2+y^2=1, y=0\}.$$ Let $R$ be the region in $\R^2$ that is enclosed by $C_1$ and $C_2$. Let $C$ be the union of the curves $C_1$ and $C_2 with a counter-clockwise orientation.
How can I determine the line integral \oint_{\partial R}\F\bullet\mathrm d\mathbf r$, where $\partial R$ is the boundary of $R$ (which consists of $C_1\cup C_2$). I feel like Green's theorem is the easiest way, since the orientation is positive. I have tried a direct computation with a parametrisation, but this makes the line integral quite difficult, due to the $\exp(y^2)\cos(y)$ in $\F$.