There is no known method for solving this integral explicitly. Your best bet is using a numerical method. Consider...
The length of an arc defined by the vector equation from t=A to t=B is
So the integral you have posed is equivalent to asking the circumference of the ellipse defined by . Take the derivatives to see what I mean. As you probably know, the circumference of an ellipse cannot be given in closed form.
You can either use an infinite series or approximation, which can be found here: Ellipse - Wikipedia, the free encyclopedia