I have an integral in the following form:
$$\int_{-\infty}^{\infty}\mathrm{d}t_1\int_{-\infty}^{\infty}\mathrm{d}t_2\int_{-\infty}^{\infty}\mathrm{d}t_3\int_{-\infty}^{\infty}\mathrm{d}t_4 f_1(t_1)f_2(t_2)f_3(t_3)f_4(t_4)\exp[-g(t_1,t_2,t_3,t_4)-\mathbb{i}c(t_1t_3-t_2t_4)]$$
where $\mathbb{i}=\sqrt{-1}$ is imaginary number, $c$ is a constant, $f_i(t_i)$, $i=1,2,3,4$, are polynomials (in fact, $f_i(t_i) = H_{n_i}(a_it_i)$ where $H_{n_i}$ are $n_i$-th order Hermite polynomials), and $g(t_1,t_2,t_3,t_4)=\sum_{i,j=1}^n b_{ij}t_it_j$ where $b_{ij}$ are the entries of symmetric matrix.
Note that the "Fourier-like terms" $\exp[-\mathbb{i}ct_1t_3]$ and $\exp[\mathbb{i}ct_2t_4]$ would be the usual Fourier transforms if, in the first case, either $f_1(t_1)$ or $f_3(t_3)$, and, in second case, either $f_1(t_1)$ or $f_3(t_3)$, would be constants. Also, unfortunately, $b_{ij}$'s don't match coefficients inside Hermite polynomials, disallowing us from using the fact that they are eigenfunctions of Fourier transform.
For polynomials of small enough degrees, this is be solvable by hand. However, the orders of my polynomials would make such calculation very tedious.
I actually know all the coefficients, and just need to numerically evaluate this integral. I am wondering if there is a better way than just using a 4-d Simpson quadrature.
To evaluate $$\int_{\mathbb{R}^n} P(x) e^{-\pi x^T M x}dx$$ where $P$ is a polynomial and $M$ a positive-definite complex matrix,
start with $$e^{-\pi x^T x} = \int_{\mathbb{R}^n} e^{2i\pi \langle \xi,x \rangle} e^{-\pi \xi^T \xi} d\xi, \qquad P(M^{-1/2} x) e^{-\pi x^T x} = \sum c_\alpha \partial^\alpha\int_{\mathbb{R}^n} e^{2i\pi \langle \xi,x \rangle} e^{-\pi \xi^T \xi} d\xi$$ $$P(x) e^{-\pi x^T M x} = \sum c_\alpha \partial^\alpha\int_{\mathbb{R}^n} e^{2i\pi \langle \xi,M^{1/2} x \rangle} e^{-\pi \xi^T \xi} d\xi$$
Where at first $M$ is real positive-definite and $\int_{\mathbb{R}^n}P(x) e^{-\pi x^T M x}dx$ is obtained by inverse Fourier-transform. Then (if you really meant $M$ complex) try extending by analytic continuation.