How can I find $\int{\sqrt{\left(b^2-1\right)x^2+1\over-x^2+1}}dx$?

Question

I got this from the perimeter of an ellipse. I came up with the formula: arclength of f(x) for x from a to b=$\int_a^b\sqrt{f'(x)^2+1}dx$. Since an ellipse has the equation: $$\left({x-h\over a}\right)^2+\left({y-k\over b}\right)2=1$$ The arclength can be found to be: $$\int_{-b}^b\sqrt{\left(b\sqrt{1-\left(x-h\over a\right)^2}+k\right)^2+1}$$ Which, if you let h and k equal 0 and a equal 1, is the problem in the title. But how do I proceed from here.