Consider an open circle $\mathcal{C}$ in the complex plane, which is centered on point $1$ of the radius $\varepsilon$ and doesn't intersect with $1-\varepsilon$ oriented in the anti-clockwise direction. Furthermore we define $f:\mathbb{C}\rightarrow\mathbb{C}$ $$f(z) = \cfrac{\log(z-1)}{z^2}\,x^z,$$ where $z\in\mathbb{C}$ and $x\in\mathbb{R}^+$.
Question: What is the exact value of $\int_\mathcal{C}f$ ?
Parameterization isn't helpful, because the integral does not tend to zero.