Peristaltic flow of a third grade fluid in a circular cylindrical tube is undertaken when the no-slip condition at the tube wall is no longer valid. The governing nonlinear equation together with nonlinear boundary conditions is solved analytically by means of the perturbation method for small values of the non-Newtonian parameter, the Debroah number. A numerical solution is also obtained for which no restriction is imposed on the non-Newtonian parameter involved in the governing equation and the boundary conditions. A comparison of the series solution and the numerical solution is presented. Furthermore, the effects of slip and non-Newtonian parameters on the axial velocity and stream function are discussed in detail. The salient features of pumping and trapping are discussed with particular focus on the effects of slip and non-Newtonian parameters. It is observed that an increase in the slip parameter decreases the peristaltic pumping rate for a given pressure rise. On the contrary, the peristaltic pumping rate increases with an increase in the slip parameter for a given pressure drop (copumping). The size of the trapped bolus decreases and finally vanishes for large values of the slip parameter.