I found the way to compute the integral. Firstly, subsitute ##y=\alpha x## to get ride of the ##\alpha## term and then factorize ##E## under the square root. Then substitute ##\sin z = \frac{U_0}{E}\tan y##. If you do everything correctly you will end up with the definite integral:
$$...
Homework Statement
Determine the period of oscillation, as a function of the energy, when a particle of mass ##m## moves in a field for which the potential energy is
$$ U = U_0 \tan^2\alpha x.$$
Homework Equations
The relevant equation is given by the general formula for the period...