The problem of inversion of the integral Laplace transform, which belongs to the class ofill-posed problems, is considered. Integral equations are reduced to ill-conditioned systemsof linear algebraic equations, in which the unknowns are either the coefficients of the seriesexpansion in terms of special functions, or the approximate values of the desired originalat a number of points. Described method of inversion using special quadrature formulas ofthe highest degree of accuracy and the characteristics of the accuracy and stability of thismethod are indicated. Quadrature inversion formulas are constructed, which are adaptedfor the inversion of long-term and slow processes of linear viscoelasticity. A method of de-formation oftheintegrationcontourintheRiemann—Mellininversionformulaisproposed,which leads the problem to the calculation of certain integrals and allows obtaining errorestimates. A method is described for determining the possible breakpoints of the originaland calculating the magnitude of the jump at these points.