Mutlivariate Gaussian integral with Covariance Matrix depending on the integration variable

Question

I have to solve the following integral:

$\int d^nx \exp(-\frac{1}{2} \mathbf{x}^\intercal C^{-1}(\mathbf{x})\mathbf{x}) \frac{1}{|2 \pi C(\mathbf{x})|} $

but as you can see, the covariance matrix also depends on my integration variable. Are there any known ways to treat this kind expression?