This paper addresses the problem of the development of a fast numerical model for the computation of the electromagnetic field in the quasi-static regime. A semi-analytical approach is achieved in order to obtain fast simulation tools dedicated to the simulation of complex configurations of eddy current (EC) inspection: an EC probe scans a conducting cylindrical tube of complex shape and with varying electromagnetic properties. In this paper, though the cylindrical tube to be inspected presents some angular symmetry, the complexity of the configuration lies in the arbitrary shape of the tube's walls and the dependence of constitutive parameters of the material on the radial or axial coordinates. In this context, no modal approach can be used to expand the components of the fields. This problem is overcome by using a pseudospectral Fourier method. Maxwell's equations are written in a covariant form in order to translate boundary conditions at each interface in an analytical form. Tangential components of the fields with respect to the boundary surface are expanded by using a high order Chebyshev polynomials. Spatial derivatives of the components of the fields with respect to the radial coordinate are thus approximated at some collocation points. Numerical validations are discussed in order to show the efficiency of the proposed numerical model.