The expansion of the electronic density and the exchange-correlation potential in auxiliary functions is a successful technique to reduce the computational effort in linear-combination-of-Gaussian-type-orbitals density functional theory (LCGTO-DFT) methods. A new approach for the efficient calculation of the necessary three-center overlap and electron repulsion integrals for this technique is presented. A new set of recurrence relations is derived, which take advantage of the special structure of the auxiliary function sets (primitive functions with shared exponents). Pathway diagrams from uncontracted integrals over s functions to any given class of target integrals are presented. The efficiency of different paths is discussed on the basis of floating-point operations (FLOPs). The FLOP counting indicates that the new method represents a substantial improvement for the calculation of three-center overlap and electron repulsion integrals.