To compute long term integrations for the pantograph differential equation with proportional delay qt, 0 < q ≤ 1: y'(t) = ay(t) + by(qt) + f(t), y(0) = y0, we offer two kinds of numerical methods using special mesh distributions, that is, a rational approximant with ‘quasi-uniform meshes’ (see E. Ishiwata and Y. Muroya [Appl. Math. Comput., 2007, 187: 741-747]) and a Gauss collocation method with ‘quasi-constrained meshes’. If we apply these meshes to rational approximant and Gauss collocation method, respectively, then we obtain useful numerical methods of order p? = 2m for computing long term integrations. Numerical investigations for these methods are also presented.