Building on Gianluca's response, one can get s result in several minutes by providing plausible assumptions.
Integrate[(r^2 - 2 r (a Cos[\[Theta]] - b Sin[\[Theta]]) + a^2 +
b^2)^(-1), {r, 0, R}, {\[Theta], 0, \[Pi]},
Assumptions -> {0 < R, Element[{a, b}, Reals], a^2 + b^2 > R^2 > 0}]
This will take several minutes and it gives a result longer than I care to post.
