We have used computer algebra (CA) to construct general formulas for the two-center overlap, resonance, Coulomb and hybrid integrals over Slater orbitals (STOs). Individual formulas then were produced for the Coulomb integrals containing all combinations of K, L and M shell orbitals. The numerical evaluation of these formulas is very rapid and allows unrestricted accuracy. Tests give complete agreement with the output of numerical programs written by other workers. Computer algebra eliminates terms of equal magnitude and opposite sign that can cause catastrophic loss of accuracy in a purely numerical computation. Strong parallelization is possible in the construction and evaluation of the formulas. Their availability opens up the prospect of carrying symbolic computation into the eigenvalue finding stage of quantum chemistry. The present calculations also highlight several needs and opportunities in the field of computer algebra.