No output from Integrate

Questions
1

I know it’s doing something because it uses a bunch of system resources for a few hours than just stops. Anyone know what I’m doing wrong?

original Input:

Integrate[(r^2 - 2 r (a*cos (\[Theta]) - b*sin (\[Theta])) + a^2 + b^2)^-1, {r, 0, R}, {\[Theta], 0, \[Pi]}]

Screenshot2024-01-08165551.png
Screenshot2024-01-08165551.png860×882 20.5 KB

2

You must first have a look at the very basics of Mathematica syntax: capitalization of commands and square brackets for the arguments of functions. Done that, your integral seems very complicated because of the three symbolic parameters. You can do special cases:

With[{a = 1, b = 0},
 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 < Sqrt[a^2 + b^2]]]
With[{a = 1, b = 0},
 Integrate[(r^2 - 2 r (a Cos[\[Theta]] - b Sin[\[Theta]]) +
     a^2 + b^2)^-1,
  {r, 0, Sqrt[a^2 + b^2]}, {\[Theta], 0, \[Pi]}]]
3

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.