I'm not sure how much more "closed form" you need than elliptic integrals. For another approach, expand sqrt(1 + B'(t)^2) in a Chebyshev series and you're off

I'm not sure how much more "closed form" you need than elliptic integrals. For another approach, expand sqrt(1 + B'(t)^2) in a Chebyshev series and you're off to the races.