Proof that doesn' t exists $f\in \mathcal{O}(\Theta)$ such that $\frac{df}{dz}=g(z)$, for all $z\in\Theta$.

Question

Let $\Theta=\{z\in \mathbb{C}\mid 1<|z|<2\}$ and complex function $g(z)=\frac{(e^{z}-z-1)^{9}}{(1-\cos{z})^{8}(e^{z}-1)^{3}}$.
Proof that doesn' t exists $f\in \mathcal{O}(\Theta)$ such that $\frac{df}{dz}=g(z)$, for all $z\in\Theta$.
We observe that $g$ has pole at $z=0$ with order 1. My first thought was to use Residue Theorem, for proving that doesn't exist $f$, but $\Theta$ is not a simply connected set. How can we proof this claim ?