I think about this phrase constantly: "The secret is not to care." Because if I don't want to let certain things make me unhappy, the secret is not to care. (Not to mention not caring about the weird grammar of the phrase.)
Compare these two sentences:
- The secret is not to care.
- The secret is to not care.
The words are the same--but their meanings are different. The first doesn't say what the secret is. It just says that not caring isn't the secret. Anything could be the secret. The secret is not to care; it's to wear plaid pants.
The second is clearer. The secret is to "not care." You'll be a happier person, in other words, if you don't give a darn. This is probably true. And yet we find ourselves caring more about words than we do our own happiness. Alas!
This whole business of split infinitives is maddening. Despite what you might have heard from your English teachers, there is NO rule in English that requires the to+verb of an infinitive to appear together. It was an attempt long ago to make English more like Latin, where the infinitive form is one word and can't be split. This is worse than the rule banning white shoes after Labor Day, because at least that rule spares us from the sight of ugly footwear.