A nonsingular form of the point kernel integrand based on the work of Ono and Tsuro has been implemented for cylindrical sources in two point-kernel computer programs. Numeric integration over the source volume is accomplished by Gauss quadrature or by a three-dimensional extension of Patterson's automatic quadrature algorithm. The latter flux calculation method is two to five times faster for exterior detector points when compared to the Patterson algorithm quadrature of the conventional form of the point kernel over a cylinder.