@Alexandre Eremenko For n = 1 and n = 2, I could evaluate it and the answer is finite, i.e. the integral is not divergent! If it helps you, think of it in a distributional sense, and regularize it by adding in the exponential $-\eta_k x_k^2$ and in the end take $\eta_k$ to $0+$.

@Marcel If it's not of research level, what is the answer?!

Thank you very much! Your idea works just fine. I would very much appreciate it if you could prove the second part, namely ${\rm R}_s {\rm R}_{s^{\prime}} = {\rm R}_{ss^{\prime}}$ if ${\lambda}_s {\lambda}_{s^{\prime}} = {\lambda}_{ss^{\prime}}$ starting from the definition of ${\rm R}_s$ transformation, namely $e^{-{\cal H}^{\prime}[\lambda]} = \ldots$.