Unnecessary additional information: I'm an AE novice, but this seems like it should be a really simple thing to do.
It is. You just have to learn it. Sorry, there are no easy answers. Good keyframing skills don't come over night and you have to spend some time with this stuff before it clicks.
- Clear away any rotation animation you've got on that cog.
- Set your rotation keyframes (eg. at frame 0, set the rotation to 3600˚ and at frame 250 set the rotation to 0˚).
- Click on your last keyframe and press F9. (That's "easy ease" and adds an "ease" to the keyframe, so the motion changes gradually/more naturally. I use it all the time, then tweak the motion in the graph editor if necessary).
- Job done.
If you want to refine the motion, go in to the graph editor, right click in the graph editor, and set it to "Edit Speed Graph". Tweak the handles of the keyframes to get the result you want - a horizontal line is a constant speed, and a steeply sloping (either up or down) is a rapid change of speed. (Obviously this depends on how much you are zoomed in to the graph, too!).
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You can also right click on your keyframe and select keyframe interpellation. From the dialogue you can set the influence of the entry and exit to the keyframe. Experiment with those settings using 0% and 100% in both fields.