Arcana Reforged quest

[Khazidhea]

I've just arrived at Winterhelm and got the quest to find the book Arcana Reforged. Now my problem is that I've already found the book, have it in my inventory, yet find no way to give the book to the quest giver. On the map, the quest arrow points to the chest where I found it, yet I've gone back and it's still empty. Any help would be appreciated.