An approximate calculation of Hadamard integrals of a special form

In the problem of distinguishing between the center and the focus for the case of a complex monodromic singular point under certain general conditions, the coefficients of the asymptotic representation of the monodromy transformation are expressed in terms of the Hadamard integrals of the functions arising from the systems of equations in variations corresponding to the edges of the Newton diagram. The integrands in these integrals are very complex, and even numerical finding the integrals from them is almost impossible. A method is proposed for calculating this kind of Hadamard integrals by solving some special systems of differential equations. The algorithm is implemented in the Maple system. A numerical example is given.