A (contour?) integration (even if by using Mathematica!)

I need to be able to calculate integrals of this type where the sum over $R$ is the sum over representations of a Lie group $G$ on whom $dU$ is the Haar measure and $\chi _ R ()$ is the character of the argument in representation $R$ ,

$$\int [dU] \exp \left\{ \sum _R \sum _ {m=1} ^{m=\infty} \frac{1}{m} f(x,y) \chi _R (U^m) \right\}$$

For a particular case I want to integrate,

$$\int _ {-\pi} ^ {\pi} \prod _{i=1} ^{N_c} d\alpha_i \prod _{i < j} \sin^2((\alpha_i - \alpha_j)/2) \exp \left\{ \sum _ {n=1} ^ {\infty} \frac{1}{n}\frac{2}{x^{-n/2} + x^{n/2}} \sum _ {i,j = 1}^{N_c} \cos(n(\alpha_i - \alpha_j)) \right\}$$

• I have little hopes that this can be done by hand but is there a way that it can be done by the computer?

• Can the above be recast as a contour integral in $z_i = e^{i\alpha_i}$ and then is this somehow picking put the constant term? Is there such an interpretation?

• I have feeling that the stuff in the exponent is divergent so one has to anyway remove those divergent pieces to be able to make sense of it - are there known techniques of handling this kind of an expression?

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Your accept rate hurts... –  DonAntonio Oct 17 '12 at 4:30
@DonAntonio I can see your observation, May be you can look through my questions and suggest if there is any answer which I should have accepted but I didn't. I see most of my questions being unanswered and only some of the answers that have come are really complete or very useful. If I have missed any then may be you can point out. –  user6818 Oct 24 '12 at 16:32