For an integer $k\geq 1$, let $f:\mathbb{C}^k\to\mathbb{C}$ be such that $f$ is analytic in the region $\text{Re}(u_i) > -1$ (say) for each $1\leq i \leq k$, and decays rapidly on vertical lines (i.e. rapidly enough to justify whatever manipulations we need for shifting contours). Let $X$ be a large real number and consider the $k$-fold complex integral $$ I_k = \frac{1}{(2\pi i)^k} \int\limits_{(1)} \cdots \int\limits_{(1)} f(u_1,\ldots,u_k) X^{u_1+\cdots+u_k}\prod_{1\leq i \leq j\leq k} (u_i+u_j)^{-1} du_1\ \cdots du_k. $$ Here the integrals are over the vertical line $[1-i\infty,1+i\infty]$.
By shifting the lines of integration to the left, one should have (based on calculations for some small values of $k$) $I_k = P_k(\log X) + O_f(X^{-1+\varepsilon})$, where $P_k$ is a polynomial of degree $\frac{k(k-1)}{2}$ with coefficients depending on $f$ and its derivatives evaluated when all the $u_i = 0$. The determination of this polynomial, however becomes difficult quickly once $k$ is even moderately large, say $k\geq 3$, since the product contains $\frac{k(k+1)}{2} \asymp k^2$ terms.
My question is, has someone already proved identities of this form for these kinds of integrals? Basically, I would like to save myself the effort of calculating these explicitly for some certain cases, but I'm not sure how to search in the literature to find references for these, if they exist. Any help or feedback would be most helpful.
