I have the characteristic function (CF) of a random variable $X$, $\psi_X(t)$, and I want to find the cumulative distribution function (CDF) of that random variable from its CF. I know that I can find the PDF directly from the CF by taking the inverse Fourier transform of the CF, and I can then integrate the PDF to get the CDF. But, I am looking for a more direct way to get the CDF from the CF, because in my case the integration to get the PDF, then the CDF are both numerical, and I want to reduce the numerical integrations from 2 to 1.
Is there a formula to derive the CDF from CF directly?