Analytic matrix elements for the two-electron atomic basis with logarithmic terms

The two-electron problem for the helium-like atom/ions in $S$-state is considered. The basis containing the integer powers of $\ln r$, where $r$ is a radial variable of the Fock expansion, is studied. In this basis, the analytic expressions for the matrix elements of the corresponding Hamiltonian are presented. These expressions include only special functions presented by the built-in $Mathematica$ codes, what enables very fast and accurate computation of the matrix elements. The decisive contribution of the correct logarithmic terms to the behavior of the two-electron wave function in the vicinity of the triple-coalescence point is reaffirmed.