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One-electron integrals over three centers and two-electron integrals over two centers, involving Slater-type orbitals (STOs), can be evaluated using either an infinite expansion for 1/r12 within an ellipsoidal-coordinate system or by employing a one-center expansion in spherical-harmonic and zeta-function products. It is shown that the convergence characteristics of both methods are complimentary and that they must both be used if STOs are to be used as basis functions in ab initio calculations. To date, reports dealing with STO integration strategies have dealt exclusively with one method or the other. While the ellipsoidal method is faster, it does not always converge to a satisfactory degree of precision. The zeta-function method, however, offers reliability at the expense of speed. Both procedures are described and the results of some sample calculation presented. Possible applications for the procedures are also discussed.