One body problem: position as function of time

Question

Consider the typical one body problem (e.g. earth-sun system), where the orbit is elliptical. It is known that there is no "closed-form" formula for the position of the point (earth) as a function of time. This is because the calculation of the position in terms of time, involves the solution of "Kepler's equation": $u-e \ sin(u)=\zeta$, where $e$ is the eccentricity (see p. 62 of "Mathematical Aspects of Classical and Celestial Mechanics" by Arnold, Kozlov, Neishtadt). However, in Spivak's book on Mechanics (p.73), he says (without justification) that the problem is elliptic integrals. This is also claimed here. I cannot see the connection with elliptic integrals. Do they get involved in solving Kepler's equation in some way?