There actually is an analytical solution using a power series that actually converges (Karl Sundman's work). Unfortunately, the universe still mocks our attempts. Though the series converges, it does so incredibly slowly. From Wikipedia:

The corresponding series converges extremely slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^8000000 terms.