1. Carefully check your + and - signs in the original problem. You've currently an unfactorable polynomial in the numerator with a larger, higher degree polynomial in the denominator. I'd agree with
@myininaya's approach. One missed sign could make it factorisable such that something could simplify/cancel out and the whole thing become much more feasible.
2. There isn't an easier way, that I can see either, you're just going to have to solve that system of equations, namely you're going to have to use both methods of combining systems of equations and making substitutions, AND using known vertical asymptote values for the denominator (in this case +1 and 2i). Yes, I'm serious, use an imaginary root here, you kind of have to. Post your system of equation here, and try those roots, see what happens (things cancel out).
3. Using that I got:
A = 0
B = 1
C = 0
D = -1
E = 1
F = 1
You'll also need to make use of this fact:
\[\frac{d}{dx}(\frac{1}{a} \tan^{-1}(\frac{x}{a})) = \frac{1}{x^2+a^2}\]
This problem as written is do-able, but it's a lot of work and will also most likely also force you to remember some anti-derivatives which have trig function forms and make use of half-angle and double-angle formulas too. So you got a lot of work to do, better get to it eh? ;-D