<p>A method for the direct integration of the three-dimensional Faddeev equations with respect to the breakup T-matrix in momentum space for three-body systems with differing masses is presented. The Faddeev equations are explicitly formulated without imposing symmetry or antisymmetry requirements on the two-body t-matrices, thus accounting for mass differences between the three interacting particles. An algorithm for the algebraic determination of non-relativistic wave functions for three-body systems with arbitrary masses is given. Furthermore, it is directly demonstrated how the domain of logarithmic singularities in the integral kernels of the Faddeev equations is significantly altered by varying the masses of the interacting particles. The developed method for traversing logarithmic singularities is tested using the example of calculating the total cross sections for elastic neutron-deuteron scattering and breakup reaction.</p>