This example outlines an aplication of the Maple + GrTensorII enviroment for the algebraic manipulation of the Dirac equation on curved spacetimes. In this purpose we used a strategy described in the articles quoted below. Mainly we first defined the Pauli and Dirac matrices algebra (using only the MAPLE environment) and then we introduced the Dirac equation on curved spacetimes using the capabilities of GRTensorII package. There are two examples of worksheets presented below, one for calculating Dirac equation on a Taub-Nut spacetime, the other one being dedicated to the Dirac equation for the Schwarzschild metric in a spacetime with torsion. Other examples (and their respective worksheets and metric files can be provided by the author, by request).