I'm not sure how you are analytically calculating path length. It seems from your expression that you just integrated r(t) straight up, which is not the correct way to do it. You need to use the formula for path length.
For example, it is given for cylindrical coordinates here:
http://tutorial.math.lamar.edu/Classes/CalcII/PolarArcLength.aspx (Google: arc length cylindrical coordinates)
That equation is difficult to integrate by hand, but it can be done symbolically in MATLAB very easily.
syms r t
r = 1 ./ ( 1 - t / (2*pi) );
ds_dt = sqrt(r^2 + diff(r)^2);
path_length = int(ds_dt,0,pi)
subs(path_length)
You get a long analytical expression for path length, which turns out to be the same answer as you got numerically, 4.47.
