"work out" without an object is to do gym exercises, as Matthew said. "work out" + object is an informal word for calculate e.g. "I need to work out the answer to this complex problem. Can you help me?" "work up" + object is not that common and is informal. It means produce by means of hard work. e.g. "I really worked up a sweat during my exercise routine."

You might also find "workout" or "work out": the former functions as a noun and it means "an exercise season" like in moving all the furnitures to the new flat was a heavy workout". The latter is a verb and it means both "to exercise" and also "to calculate" as Michael said: "I need to work out more because I'm getting fat"

To "work out" is just another way of saying "to exercise". To "work oneself up" means to make yourself excited, angry, emotional. I'm not sure what "smth" means by the way.