What is the arc-length parameterization of an ellipse?

I know the typical parameterization of an ellipse is (a cos(t), b sin(t)). However, this is not an arc-length parameterization and I can't make the usual techniques for getting an arc-length parameterization to work since there is square root in the integrand that doesn't simplify if a is not equal to b. Is there a formula for an arc-length parameterization of an ellipse?