Explicit solution of $\kappa^2u''-f(u)=0$ with certain boundary conditions

Question

Let $f(x) = -x+\frac18\ln\frac{1+x}{1-x}$ and $F(x)$ be its primitive, i.e., $$ F(x) = -\frac12 x^2 +\frac18 [(1+x)\ln(1+x)+(1-x)\ln(1-x)]. $$ Let $M\in (0,1)$ be the unique zero point of $f(x)$. I wonder if it is possible to derive the explicit solution of the following problem $$ \begin{cases} \kappa^2 u'' - f(u) = 0, \\ u(0)=u'(\frac{\pi}{2})=0,\ u(\frac{\pi}{2})=M, \end{cases} $$ where $\kappa>0$ is a constant. Multiplying the equation by $u'$ and integrating from $x$ to $\frac{\pi}{2}$, we see $$ u'(x) = \frac{\sqrt2}{\kappa} \sqrt{F(u(x) - F(M))}. $$ By separation of variables and integrating, we get $$ \int_0^u \frac{dw}{\sqrt{F(w)-F(M)}} = \int_0^x \frac{\sqrt2}{\kappa} ds = \frac{\sqrt2}{\kappa}x. $$ However, it seems hard to compute the left-hand side integration. Does anyone know how to deal with this kind of integration or any reference about this?