I tried to do a arc length integral s, for y as an elliptic function of x. But as I continued with the integration, I found myself at the above integral(cosine of 2x). I quickly substituted cos(2x) with A and carried on but got stuck after about a step or two. The new problem now became (A^0.5)/(1-A^2)^0.5 I tried integration by parts, a lot of substitutions, and nothing worked. Then I thought I should give this to wolframalpha. It gave the results but in, what I only recently found out, elliptic functions. So, does this mean there are no closed solutions/expressions to integrals like these? And does that mean there is no exact formula for applied mathematical problems involving these, like here the perimeter of an ellipse?