Computing some classes of Cauchy type singular integrals with Mathematica software. We construct an algorithm, [SInt], for computing some classes of Cauchy type singular integrals on the unit circle. The design of [SInt] is focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithm. Furthermore, we show how the factorization algorithm described in [A. C. Conceição et al., in: Operator Theory: Advances and Applications 202, 87–109 (2010; Zbl 1208.47017)] allows us to construct and implement the [SIntAFact] algorithm for calculating several interesting singular integrals that cannot be computed by [SInt]. All the above techniques were implemented using the symbolic computation capabilities of the computer algebra system Mathematica. The corresponding source code of [SInt] is made available in this paper. Several examples of nontrivial singular integrals computed with both algorithms are presented