I wonder if the following integral has an analytical solution. $$ \int_{-\infty}^{\infty}\frac{w_1 N_1(x)}{\sum_{i=1}^{n} w_i N_i(x)} N_0(x) dx $$ where $w_1, \ldots, w_n$ are positive constants, $N_0, N_1, \ldots, N_n$ are Gaussian functions with different parameters.
If no analytical solution exists, what would be the best way to efficiently approximate it? I'm considering importance sampling by sampling from $N_0$. Any suggestion of a more efficient yet not too complicated method would be appreciated.
A previous question on this site seems quite related to mine, but no solution has been given there: How to solve an integral with a Gaussian Mixture denominator?