I am trying to solve for the following unpleasant integral
$$\int_{\gamma}^{\infty} \bigg[t- \int_{-2}^{2}\frac{ t \ f_X(x)}{1+N \ \big|G(x)\big|^2 \ t^{-3}}\ dx\bigg] \ dt$$
where $N$ is a non-negative integer for example 16 and where
\begin{align} G(x) = \frac{1}{N} \text{exp} \bigg( j \pi x \frac{(N -1)}{2}\bigg) \frac{\sin (\pi \frac{N}{2} x)}{\sin (\frac{\pi}{2} x )} \end{align} G is a sinc function.
And where $$f_X(x) =\int_{\max\{-1,-1-x\}}^{\min\{1,1-x\}} \left(\frac{1}{\pi^2 \sqrt{1-(x+y)^2}}\frac{1}{\sqrt{1-y^2}}\right) dy $$
Solution
The way I do this is
1- Using MATLAB I first numerically integrate $f_X(x)$ i.e integrate over $y$ and keep it in terms of x
2- then I do Reimann sum for my inner integral: $$x=-2:0.1:2$$
$$\sum_i \frac{ t \, \ f_X(x_i)}{1+N \ t^{-3} \ |G(x_i)|^2} \times 0.1$$
3- I symbolically integrate for t
Question
Do you have any nice smart tricks to simplify? or any approximation?