I am trying to find analytic solutions of this equation for $x$ with parameter $a$ ($x>0, a>0$):
$$(\log(ax))^2\frac{x}{(x+1)^2}+2\log(ax)\frac{1}{x+1}+2\log\left(1+\frac{1}{x}\right)-2\log(1+a)=0$$
From simulations I see that the equation has two solutions. One is easy to find (just by plugging in) $$x_1=\frac{1}{a}$$ I'm unable to find the second one. Is their a closed-form solution for $x_2$?