There are many ways to solve potential problems for anisotropic mixtures. The finite element method is a popular choice with great flexibility that gives the solution in the entire computational domain. Effective medium approximation is a simpler option that only estimates effective properties. Interface integral equation methods is a third class of methods that has been studied for a long time, but only rarely been used for computations. In this paper, electrostatics of anisotropic two-dimensional composites is discussed. The geometry under consideration is a periodic composite of arbitrarily shaped anisotropic inclusions in an anisotropic matrix. The location of the inclusion interfaces are given on analytic form as to facilitate reproduction of results. Highly accurate calculations are performed with a coupled first and second kind Fredholm integral equation method. Comparison is made with finite element method calculations.