Consider the contour integral

$\frac{1}{2\pi i}\oint_\gamma\chi(z)\frac{dz}{z}\,,$

where $\gamma$ is a (not necessarily simple) closed curve lying in $\mathbb{C}\setminus{0}$ and $\chi\colon\mathbb{C}\to\mathbb{R}_{\ge 0}$ is a continuous function. My question is

Are there special/generic hypotheses on $\chi$ that allow for closed-form expressions of the contour integral above?

Of course, in the trivial case of a constant function $\chi$, the integral is simply the constant time the winding $n(\gamma,0)$ of $\gamma$ about the origin. I suspect that the integral will be a (non-negative real) multiple of $n(\gamma,0)$ but I am unable to figure out how to proceed or determine this factor, if so.

I am particularly interested in finding a closed form expression when $\chi(z)=\langle z\rangle^{-2}$, where $\langle\cdot\rangle:=(1+|\cdot|^2)^{1/2}$ is the Japanese bracket, which has arisen in a harmonic analysis context. If this specific case is known in the literature or tractable, I would be glad for a reference or a proof.