Evaluate $\int\cdots\int_{\mathbb R^n}\prod_{i=1}^{n}\frac{x_i^me^{-x_i^2}}{e^{\frac{x_i}2}\cosh(\frac{x_i}2)}\mathbb dx_1 \cdots \mathbb dx_n$

Question

I'm wondering how can we evaluate $$\int\cdots\int_{\mathbb R^n}\prod_{i=1}^{n}\frac{x_i^me^{-x_i^2}}{e^{\frac{x_i}2}\cosh(\frac{x_i}2)}\mathbb dx_1 \cdots \mathbb dx_n$$ using Gamma/Beta functions? Thanks in advance.