The attached document is an example of the Legendre Quadrature
applied to the popular 1/(1+x) extreme case. Two sets of coefficients
are given in the collapsed area [63, 64]. Adapt your difficult project
from this model.
"This page was compiled by Mike "Pomax" Kamermans, on June 5th, 2011,
as a resource for the primer on Bezier curves, specifically the section
that deals with arc length computation. You are free to do what you like
with this data. Go make the world a better place than it was when you found it."

https://pomax.github.io/bezierinfo/legendre-gauss.html

JeanIntegrate Quadratures Legendre EXPRESS.sm (56 KiB) downloaded 95 time(s).