The anime is called "The Melancholy of Suzumiya Haruhi" (涼宮ハルヒの憂鬱 Suzumiya Haruhi no Yuutsu), it's an anime about, uh, espers, aliens and time travellers. Kind of hard to explain, just google it. It was quite popular back when it was aired in 2006 and when it was "rereleased" in 2009 with additional episodes.

This clip is the "deduction" scene from episode 8 (in the original 14 episodes run), with music from the game Phoenix Wright. (Just to be clear, the rest of the anime has nothing to do with Phoenix Wright/Ace Attorney.)

The PW/AA music is just added on top of the original music, since I don't have the expertise to remove the original music without removing the voices. But the original music is so low, it's barely audible, so I just left it like this. Also added a few sound effects from the game.