I am looking for a periodic solution (with period $2\pi m$, for some integer $m$) to the following ODE $$\ddot{x}+x-\frac{1}{x}=0.$$ ($x(t)=\pm 1$ trivially solves this equation). Note that this system is integrable, so the quantity $$E=\dot{x}^2+x^2-\log x^2$$ is constant for any solution. Separating variables and integrating gives $$t=\int\frac{dx}{\sqrt{E-x^2+\log x^2}}.$$ However, I haven't been able to solve the integral. Maybe thinking of this as a physical system with potential $V(x)=x^2-\log x^2$ would be useful (though I haven't been able to employ this fact).
I'd appreciate any suggestions.
It is easy to conclude the existence of closed orbits due to the energy formula structure.
The energy level contour lines are given by
$$ \dot x^2+x^2-\log(x^2) = E $$
Near the minimum they have the aspect
In red is shown the level $E = 4$ and in green an orbit described for $x_0 = 0.15, \dot x_0 = 0$.
Following, in red the plot for $\dot x$ and in blue the plot for $x$