This problem was found on an old post and i thought it was an interesting problem. a hint was given which said remember that tan x = cot(pi/2 - x). i tried letting u = pi/2 - x and i got:

\(\displaystyle \int_0^{\pi/2} \frac{du}{1+(\cot u)^{\sqrt{2}}}\)

i'm not sure what to do next though. my substitution didn't seem to do much.