Evaluating $\int \frac{sin^2(x)}{\sqrt{cos(x)}} \mathrm dx$

I would like to get some advice how to evaluate the integral,

$$\int \frac{\sin^2{x}}{\sqrt{\cos{x}}} \mathrm dx$$

• This question is not trivial! – freak_warrior Nov 16 '13 at 8:22
• not at all trivial : goo.gl/FwZsTp – lab bhattacharjee Nov 16 '13 at 9:04
• Integrals are not solved, they are evaluated, computed, ... To solve, need to say solve it what. – Arjang Nov 16 '13 at 9:33

Integration by parts, with $g(x)=\sin x$, and $$f'(x)=\frac{\sin x}{\sqrt{\cos x}}=-2\cdot\frac{\cos'x}{2\cdot\sqrt{\cos x}}=-2\cdot(\sqrt{\cos x})'\iff f(x)=-2\cdot\sqrt{\cos x}$$ then recognizing the expression of the incomplete elliptic integral of the first kind in $\int f(x)g'(x)dx$