some algebraic curves:
parametrizations, arc lengths, etc.:

circle
$\cos,\sin$ or by the algebraic one
$\frac{t}{1-t^2},\frac{1}{1+t^2}$ or some such thing...
  (use the $u=\tan t/2$ substitution)
arc length: easy

ellipse
$a\cos t, b\sin t$
(similar version of algebraic one)
arc length: super-hard: elliptic integral!!

parabola
arc length is related to area under hyperbola

(and guess that arc length of hyperbola is area under parabola??)

$x^2=y^3$ (parametrized by $\cos,\sin$)
arc length: pretty easy

nodal curve?

I bet that elliptic curves are super-hard to compute arc length
