In Arnold's Mathematical Methods of Classical Mechanics he uses,
$\int_0^1\frac{1}{\sqrt{x^b - x^2}}\,\text{d}x = \left\{\frac{\pi}{2-b} : 0\leqslant b < 2\right\}$ but he doesn't explain how to get it.
Via Mathematica (wolfram alpha works also), $\int\frac{1}{\sqrt{x^b - x^2}}\,\text{d}x = -\frac{2x^{b/2}\sqrt{1-x^{2-b}}\arcsin{\big(x^{(2-b)/2} }\big)}{(2-b)\sqrt{x^b-x^2}}.$
(Besides using a CAS) I am not sure how to solve the general case or the specific improper integral.