Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which $$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$ and $$x_1+\ldots+x_k\leq 1.$$ I am interested in the integral of $\frac{1}{x_1x_2\ldots x_k}$ over this region. I am also interested in generalising this slightly by replacing the second constraint with $$x_1+\ldots+nx_k\leq 1,$$ for some $n>0$.

I would like to evaluate these integrals, or give an upper bound which is of the correct order of magnitude, for $k\approx 100$. Currently, I can get an upper bound of $\delta^{-k}$ times the volume of the region, the volume can be computed by changing variables to $(x_1-\delta,x_2-x_1,\ldots,x_k-x_{k-1})$. However, I suspect that this is a considerable overestimate.

I would be interested to know if these integrals can be evaluated exactly. I believe they could be expressed in terms of polylogarithms, but even for $k=3$ the expressions appear to be very messy. If not are there any suitable numerical methods I could use?