Integral of sqrt(tan(x))

By considering:

\displaystyle \int \left(\sqrt{\tan x} \pm \sqrt{\cot x}\right) \mathrm dx

evaluate the classic integral:

\displaystyle \int \sqrt{\tan x}\mathrm dx

Hence evaluate:

\displaystyle \int \frac {x^2} {x^4 + 1} \mathrm dx