Abstract: Applying the integral Laplace transform to a wide class of problems leads to a simpler equation with respect to the image of the sought original. The next step is the inversion problem, i.e., finding the original from its image. This step is typically not analytically feasible, necessitating the use of approximate inversion methods. In this case, the approximate solution is represented as a linear combination of the image and its derivatives at a number of points in the complex half-plane in which the image is analytic. However, the original, unlike its image, may even have discontinuity points. Finding frameworks and approximate solutions is reduced to solving systems of linear algebraic equations (SLAE) constructed using classical orthogonal Laguerre, Legendre, and Chebyshev polynomials and their generalizations. SLAE matrices have various properties, which, when taken into account, allow decreasing their condition number as compared to known regularization methods. The results of numerical experiments confirming the effectiveness of the proposed inversion algorithms are presented.