What is the best and most reliable way to detect if a book is pulled out of a book case?

As Majenko stated in the comments, an easy and discrete way would be using IR proximity sensors behind each book. The IR light from them is invisible to the eye, they are easy to read with an Arduino and often the casing of the IR LED and IR sensor are black, so that you could easily hide them in a black backplate of your book case.

The IR sensor will output one pin state (HIGH or LOW depending on the sensor), when something (the book) is near it (on most sensor modules you can change to threshold to your liking), and the other, if nothing is near it (empty place in the book case)

Searching "ir proximity sensor" at amazon gives multiple good looking results for ready to use sensor boards.

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Another - more hacky - way would be to build your own switches at the top of the book case. As a switch is nothing more than 2 electrical contacts with some mechanic suroundings, you could add a small piece of cardboard above each book, which falls down a bit, when the book is removed, but not enough, that it is easily spotted (a few millimeters should be enough). At the top of the case and on the cardboard piece you put some piece of tinfoil or aluminum tape. When the book is put in the box, the tinfoil pieces get pressed together by the book, thus make contact. Wire one piece to ground, the other to an Arduino pin.

I find the IR sensor solution better, but I wanted to include a solution, where you don't need to buy extra sensor modules.