I think if you realy want to mix cyclic and normal animation, you can do that with the NLA editor but if your animation is just an object rotating then slowing down and stop, I think there is an easier way of work.
On your object create a rotation keyframe on frame 1 then go to the frame where your object is suposed to slow down, create a keyframe with as many turns as you want (for example 1080 degrees for three turns before slowing down).
Now go to the frame where your object is suposed to stop then add as many turns as you want before stoping and again create a rotation keyframe (for exemple 1440 degrees and your object will turn three time and it will slow down during one more turn then stop)
You need to set linear interpolation between the first and second keyframes and bezier between the second and third ones.
Obviously you should also need to tweak some timing, f-curves, turn numbers …etc
This is the way I would use if I had to do this animation.