I don't really have any idea about how does one calculate integrals involving Multi-valued functions, all I know about Multi - valued functions, are the closely associated concepts of Branch Points and Branch Cuts, about which almost the entirety of my knowledge can be summarised by THIS: https://math.mit.edu/classes/18.04/Notes/Branch_Points_B_Cuts.pdf
SO, I understand the basic concept behind a multi valued function ( i.e. what makes it be ambiguously defined ), how to compute these branch points in most cases and understand that generally speaking branch cuts are curves that join two of these branch points, that's about it.
I do not have a working knowledge of how to integrate these functions into the grander whole of things, which is what this problem really expects me to have.
So, I just understand that the Branch points of this function are: +1,-1 and infinity and further that, when we consider an 'appropriate branch' of the $log(z^{2}-1)$ function, we will be able to apply the Residue theorem to solve this problem since this function, when unambiguously defined is Meromorphic, but how do I go about using that knowledge to define 'branches' of this function is the step that I really require an explanation for.
