  We develope in great computational details the classical Cartan equivalence
problem for Levi-nondegenerate C^6-smooth real hypersurfaces M^3 in C^2,
performing all calculations effectively in terms of a (local) graphing function
\varphi. In particular, we present explicitly the unique (complex) essential
invariant J of the problem. Its expansion in terms of the 3-variables function
\varphi incorporates millions of differential monomials, while, when \varphi is
assumed to depend only on 2 variables (rigid case), J writes out in two lines
(7 monomials).
