Twisted winding number

Consider the contour integral

$frac{1}{2pi i}oint_gammachi(z)frac{dz}{z},,$

where $gamma$ is a (not necessarily simple) closed curve lying in $mathbb{C}setminus{0}$ and $chicolonmathbb{C}tomathbb{R}_{ge 0}$ 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 $chi(z)=langle zrangle^{-2}$, where $langlecdotrangle:=(1+|cdot|^2)^{1/2}$ 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.