Integrals and Series Related to the Surface Area of Arbitrary Ellipsoids
  Integrals related to the surface area of arbitrary ellipsoids are derived, evaluated, and compared with each other and existing integrals found in the literature. We clarify the literature on the ellipsoid area problem, which dates back to Legendre's original 1811 work. We derive and evaluate several additional integrals involving or evaluating to elliptic integrals. These new integrals are natural extensions of those found in the literature and constitute useful additions to standard integrals compendia. 
R. A. Krajcik and K. D. McLenithan