From this integral (where $K$ and $s$ are constants): $$\int_0^{\frac{\pi}{2}} {\left(K^2 {\mathrm{sin}}^2 \;t+12\;{\mathrm{cos}}^2 \;t\right)}^{\frac{s+1}{2}} \mathrm{dt}=$$
This beautiful result is obtained: $$2^s\;3^{\frac{s+1}{2}}\;\pi\; _2F_1\left(\frac{1}{2},-\frac{s+1}{2};1;1-\frac{K^2}{12}\right)$$
How was that? I know, it's a hypergeometric function, but I can't figure out how it is obtained. I'd appreciate any help (perhaps how to obtain it with MAPLE). Thanks in advance!