What is the ruling on the prayer of one who entered the mosque while the Imam was bowing in the last rak'ah, and then Satan whispered to him to break his intention when he said the takbir, and he completed the prayer?

The prayer mentioned is valid because the bowing was caught with the Imam, thus the rak'ah was caught. The words of Sheikh Ibn Uthaymeen do not apply to this case, as they are specific to someone who deliberately delayed, whereas the questioner was overcome by obsessive thoughts (waswas). The Sheikh himself did not definitively state that the rak'ah was invalid in the case he mentioned.