Assuming that you have approximated or solved the Euler's Equations for components of angular velocity along its principal axes of inertia $x$, $y$ and $z$ - i.e. in the coordinate system that is rotating with the body- in the torque free case, and that the resulting functions $\omega_x$, $\omega_y$, $\omega_z$ of time for these components are integrable, how would you than go about solving the function for position of the body in the inertial coordinate system $X$, $Y$ and $Z$? I have tried using Euler's angles and to solve for the precession, nutation and spin; however, I failed at separating the terms (splitting spins, precessions and nutations into three equations). I would sincerely appreciate if someone could present how they would solve these equations, or offer a different solution (using f.ex. quaternions or rotation matrix) that would give the solution for the rotational position of the body as a function of time (or an approximation, that is still a function of time).
I am not highly interested in computing the answer: I would really like to find the function of time, even if it is only an approximate.