Convergence accelerator methods are employed to analyze some of the most difficult three-electron integrals that arise in atomic calculations. These integrals have an explicit dependence on the interelectronic coordinates, and take the form integral r(i)(1)r(j)(2)r(k)(3)r(l)(23)r(m)(31)r(n)(12) exp((-alpha(r1)-beta(r2)-gamma(r3))dr(1)dr(2)dr(3). The focus of the present investigation are the most difficult cases of the parameter set [i, j, k, l, m, n]. Several convergence accelerator techniques are studied, and a comparison presenting the relative effectiveness of each technique is reported. When the convergence accelerator approach is combined with specialized numerical quadrature methods, we find that the overall technique yields high-precision results and is fairly efficient in terms of computational resources. Difficulties associated with the standard numerical precision loss of convergence accelerator techniques are circumvented.

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