Ask HN: Which "AI" provider would you choose?

I'm looking for an "AI provider" recommendations for physics teaching and academic research. My use case requires mathematical and physics reasoning for paper parsing and PDF synthesis to generate lecture notes, exams, and problem sets. Daily volume averages would be around 100 interactions alongside 2–3 multi-page document processing tasks.

Nice to have: native web search, prompt caching for context efficiency, strict zero-data-retention or training opt-outs, and a budget of around 20ish €/month.

I've been considering Gemini pro because of its integration with NotebooklLM and other perks... I know privacy there would be a bit lacking, though. Would Claude or ChatGPT would be a more sensible choice?

Thanks a lot in advance.