The goal of this paper is to present a method leading to three different closed-form evaluations of the moment-like improper integrals \(\begin{aligned} \int \limits _A^\infty x^m\cdot (1-\tanh (x))^n\,{\textrm{d}}x, \end{aligned}\) where \(A\ge 0\) and m, n are non-negative integers with \(n\ge 1\) . These evaluations involve multiple harmonic numbers, specific tails of series representing multiple polylogarithms, and Stirling numbers of the first kind respectively. As a by-product, we also deduce a new proof of the important identity \(\begin{aligned} {{\,\textrm{Li}\,}}_{\{1\}_j}\left( \tfrac{1}{a+1},\{1\}_{j-1}\right) =-{{\,\textrm{Li}\,}}_j\left( -\tfrac{1}{a}\right) , \end{aligned}\) where \({{\,\textrm{Li}\,}}\) denotes the multiple polylogarithm function.
