Integrate angular velocity for Newton/Euler equations for single body under force load

Hi folks, i have the following task to solve: i want to calculate the point reaction forces of a body under arbitrary load of some point action forces. The point reaction forces are described as spring reaction forces for some stiff springs at given locations which will result in forces if the body is moved/rotated.

Idea is to use an ode solver to damped equations of dynamic movement (Euler/Newton) in 3d. But at some point i will have to integrate the angular velocity omega in order to solve for the rotation angle of the body in order to get the asymptotic position and rotation of the body in order to calculate the change in spring lengths and direction in order to calculate the spring reaction forces.

How do i integrate the local angular velocity?

Regards, Johannes