The request is defined as the act of asking for something or something you asked for. An example of a request is the song someone asked a band to play.

•  Modal verbs are helping/auxiliary verbs that express ideas like ability, permission, and asking for assistance. Many modal verbs have more than one meaning.
  • They are always followed by the simple form of a verb.
  • For example: Could you help me for a minute, please?
• "Could" have the same meaning as may when making requests. It is equally polite to say “Could I leave early?” or “May I leave early?”
  • Could is the past tense of "can" and is more polite and it is used with any subject to ask for permission.
  • For example “Could I open the window?” or “Could you open the window?” are both grammatical.
• Here, from the four options, the more polite form is Could you let me take the book now? It makes a request in a polite manner.

Thus, it is concluded that Your classmate has just finished reading a book from the library that you wanted and you want him/her to give it to you. Could you let me take the book now? is the most appropriate request.