Solving the equation of a kink

Question

Good morning. I was trying to calculate soliton-type solutions (kinks) in Rajarman's book, particularly with a potential of the form $$ U(\phi) = \frac{1}{2} \phi^2(\phi^2-1)^2 $$ According to the book, to obtain this solution, one must solve the integral $$ x-x_0 = \int \frac{1}{\sqrt{2 U(\phi)}} ~d\phi = \frac{1}{\phi} + \frac{1}{2} \ln(1+\phi) - \frac{1}{2} \ln(1-\phi) $$ which I have done, however, I don't see how to solve the equation $$ x-x_0 = \frac{1}{\phi} + \frac{1}{2} \ln(1+\phi)- \frac{1}{2} \ln(1-\phi) $$ for $\phi$ and obtain the same result as the book: $$ \phi(x)= \pm\left(1+\mathrm{e}^{ \pm 2 x}\right)^{-1 / 2} $$ I would greatly appreciate it if someone could indicate how to solve said equation.