I am currently working on my big honours project and need some help on a mainly coding problem I may be having. So basically in the images attached is the Lagrangian density for my soliton that I must solve numerically, initial I thought my best approach would be to implement spectral methods using Fourier transforms as that is one of the more accurate methods of solving BVPs with smooth solutions on periodic domains. Now when discussing with my supervisor he told me I was simply overcomplicating it, because we were working in one space dimension and because of the property of solitons being constant with respect to time, we can do some “Math trickier” to reduce the second order ODE to a simple integral and then find the inverse of the function gained from that integral.
Lagrangian density and converting Euler Lagrange equation to an integral
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I solved the integral using just an ordinary integral calculator as I just wanted an idea of what this function would look like and how difficult it would be to invert and here is the result:

Integral solution
my question is how do I find the inverse of this numerically?

I am honestly just trying to figure out what the best method for this question is.