  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
electric field and electric-like self-dual background. We also devise a
prescription based on the finite-part integration of the Cauchy principal value
integral to sum and extrapolate the non-alternating divergent weak-field
expansions of the Heisenberg-Euler Lagrangians to recover the non-perturbative
Schwinger effect well into the strong field limit.
