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**shawsend** Here's what I'd do: Yeah, I'd try some things first but if I couldn't figure it out, I'd input it into Mathematica so that I could at least figure out how to arrive at the answer:

$\displaystyle \int \frac{\cos(x)}{\sqrt{\sin(2x)}}dx=-\frac{cos(x)\sqrt{\sin(2x)}}{3 \sqrt{\sin(x)}}\sum_{k=0}^{\infty}\frac{(3/4)_k (3/4)_k}{(7/4)_k}\frac{\cos^{2k}(x)}{k!}$

At this point I'd ask myself, why is Mathematica reporting this? Can it be that there is no finite antiderivative? Mathematica is sometimes wrong of course and frequently expresses answers in "uncommon" ways. If there is no finite answer, then I'd try and figure out how did Mathematica come up with that answer (which is in terms of a hypergeometric series)?