You are right about the integral. As it happens, this complicated-looking integral can be evaluated fairly easily by using complex numbers and the method of

contour integration. If you make the substitution

then the integral becomes

(**)

where the integral is taken round the unit circle C. If

then Cauchy's integral formula gives the answer as

(I have skipped all the details of the calculation). If d=5 and r=1, that becomes

fairly close to the answer that you wanted.

(**)

**Edit. **I have just seen that this integral looks confusing. The "dz" in the numerator is just the usual notation for an integral with respect to z, but in the denominator "d" is a constant and "

" means d times z squared.