Hello Miles,
I offer an elaboration of the
integration
you are asking for by means of both canonical approaches applicable to this class of problems, which are (i) partial fraction decomposition of the rational integrand, integration of the proper integral, and evaluation of its limit, and (ii) an application of the residue theorem. For the first approach, I offer to demonstrate integrating the decomposition of the integrand into (a) complex terms with single poles and (b) real rational functions with quadratic denominators (and linear nominators).
For a more elaborate proposal, please check the file [login to view URL] in folder
[login to view URL]
on my Google drive.
Regards, Peter