<p>The theory of atomic structure is conceptually built on hydrogen-like orbitals and computed using Slater or Gaussian orbitals, owing to the relative difficulty of computing integrals concerning the hydrogenic orbitals. The optimal set of hydrogenic orbitals in an atom is obtained by minimizing the energy with respect to the orbitals. The Coulomb integral is difficult to compute due to the inverse distance relationship. In this paper, we evaluate the Coulomb integral and its derivative using two expressions for the inverse distance: the Laplace expression and the Legendre expression. The two expressions for inverse distance are similar and yield different integral forms. The Laplace expression yields the Coulomb integral as a sum of hypergeometric functions while the Legendre expression yields a compact polynomial form. The derivative of the Coulomb integral (computed using both forms) with respect to the decay constant is also provided.</p>