Would prefer

I've heard of 3 ways using Would prefer but I wonder which is the best (grammatically right):

I would prefer to stay at home rather than to go out (My teacher taught me so, but he's not a native one and makes errors on occasionally)



I would prefer to stay at home rather than going out(Practical English Usage)

I would prefer to stay at home rather than go out(Practical English Usage)



Since the (Practical English Usage) hasn't made any mention to the first usage, I wonder which is the best way