Correct spelling, explanation: recuperate is the correct spelling since it comes from the Latin recuperare. Recouperate is an incorrect form, probably made because of the ou sound in French originated words. However, the correct form is recuperate, because the word is of Latin origin.

Definition of recuperate:
verb, to recover
It seems that the company will soon recuperate the foregone income.
You have some time now to get some sleep and recuperate.

Incorrect spelling, explanation: this form, recouperate, is incorrect because we spell this word without o: recuperate. Ou sound is a mistake and probably was made due to similar French pronunciation. Recuperate is the correct form and comes from Latin.