find
With some algebraic steps You obtain...
$\displaystyle \frac{1}{2\cdot x^{2} + 2 + \sqrt{4\cdot x^{4} + 4\cdot x^{2}}} = \frac{1}{2} (1 - \frac{x}{\sqrt{1+x^{2}}})$ (1)
... after that integration is a 'confortable job'...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$