Coulomb integrals express long-distance (in molecular scale) interaction between electrons in the molecule. One can split nonvanishing Coulomb integrals into two groups:
Nonclassical integrals need to be treated using usual methods of quantum chemistry: use of Gaussian basis set simplifies the evaluation considerably. The nonclassical integrals have a nice property that their number remains proportional to the system size (at least for molecules having tens of atoms).
Classical integrals are more challenging to compute: their amount increases quadratically for systems occuring in quantum chemistry. The switch to linear regime occurs only for practically macroscopic systems. Fortunately, these integrals can be computed using linearly-scaling methods developed for classical charges by astrophysicists.