I am trying to find the area over the interval I=[a,b] for a stable distribution. As you know, in general, the densities of stable laws do not have explicit expressions via elementary functions. Therefore, finding the area over an integral I might be difficult. Is there a trick or a way to overcome this difficulty?

I understand that properties of stable distributions are investigated indirectly, often by using the integral transformations such as Laplacian and Fourier transformation. Can someone refer me to papers or books that answer my question?