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...
r_5 = a*(1 - (e + (1 - e^2)*arctanh(e))/(2*E(e)))
...
Average r_5 is the average distance of the orbit from the Sun where the
average is weighted by the *arc length* of the orbital path. In this
last case the arctanh(...) function is the inverse hyperbolic tangent
function and the E(...) is the complete elliptic integral of the 2nd
kind. This is the only one of these averages that differs from a by a
term of order e. All the other averages listed only differ from a by
terms of at least order e^2.