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An improved testing procedure using the marching-on-in-order method to solve the time-domain electric field integral equation (TD-EFIE) for conducting objects using the Laguerre polynomials is presented. Exact temporal testing is performed before the spatial testing, therefore the retarded terms composed of the spatial and the temporal variables can be analytically separated. The uniqueness of this testing procedure is that the time variable can be analytically integrated out and the accuracy can be improved. This paper is then an improvement over the earlier marching-on-in-order method. In addition, this methodology is quite different from the conventional marching-on-in-time algorithm as the present method leads to a set of final equations which need to be numerically solved containing only the spatial variables. Therefore, there is no requirement to have a Courant stability condition in this procedure. How the singular integrals are treated is also discussed. Several examples are simulated both for radiation and scattering problem. The results are compared with the inverse discrete Fourier transform of the frequency domain data and they agree well.