About doing an integral

Question

For a given integer $N$ and some function $f(x,y)$ I want to be able to do an integration of the kind,

$\int_{0} ^{\infty} \prod_ {i=1}^N da_i e^{[\sum_{n=1}^{\infty} \frac{1}{n} \left[ N + f(x^n,y^n)\sum _{i,j=1}^N cos(n(a_i-a_j)) \right ] ] }$

I guess that I have to anyway remove the N/n term in the exponent for anything to converge. A typical $f$ will be like $f(x,y) = -1 + 2\frac{(\sqrt{y} - 1/\sqrt{y})(1/\sqrt{x} - \sqrt{x})}{1/x - x}$.

I expect the answer to be a power-series in $x$ and $y$.

For even N=2 I couldn't manage to get an answer from Mathematica! I tried cutting off the sum over $n$ to some finite number of terms rather than an infinite sum and still I couldn't get an answer.

The code is being pasted below,

Integrate[
  Exp[Sum[
     ((-1 + (2 (x^(-n/2) + x^(n/2)) (y^(-n/2) + y^(n/2)))/(x^(-n) - x^n))
    * (2 + 2 Cos[(a1 - a2) n])
     )/n, 
     {n, 1, 2}
  ]],
  {a1, 0, Infinity}, {a2, 0, Infinity}
]

For larger N I don't know how to make it on its own generate that double sum over cosine for a list of integration variables. For small N I can may be manually type the expression but that can't be a sustainable solution.

I would like to know how to do this integral.