The subsidiary integral W (l, m, n; α, β, γ) is a key integral that appears in the variational calculation of a three-electron atomic system using Hylleraas coordinates. For the case where the ratio α/(α + β + γ) ∼ 1, an important special situation that may occur in the evaluation of the Bethe logarithm, existing approaches for calculating the W integral become impractical due to the problem of slow convergence. In this article, we present a computationally efficient and numerical stable method, in which the W integral can be expressed in terms of either a finite series or a finite recursion relation. Numerical tests are given. © 2012 Wiley Periodicals, Inc.