  We applied the method of finite-part integration [Galapon E.A Proc.R.Soc A
473, 20160567(2017)] to evaluate in closed-form the exact one-loop integral
representations of the Heisenberg-Euler Lagrangian from QED for a constant
magnetic field and magnetic-like self-dual background. We also devise a
prescription based on the finite-part integration of a generalized Stieltjes
integral to sum and extrapolate to the strong-field regime the alternating
divergent weak-field expansions of the Heisenberg-Euler Lagrangians.
