Is it permissible for a Muslim to supplicate for a polytheist and to perform ruqyah on him with the Quran and legitimate ruqyah with the intention of inviting him to Islam, and is it permissible to ask them to say, "There is no power or strength except with Allah," to affirm the doctrine of Tawhid?

It is permissible for a Muslim to perform ruqyah (exorcism/healing through Quranic recitation) on a non-Muslim, especially if they hope to invite them to Islam. This is supported by what Kharijah ibn al-Salt narrated from his uncle, who performed ruqyah on a madman with Al-Fatihah, and the man was cured, and he took a fee for it. The Kuwaiti Fiqh Encyclopedia stated that there is no disagreement among jurists regarding the permissibility of a Muslim performing ruqyah on a disbeliever. Ibn Al-Qayyim stated that Al-Fatihah is a cure even if the person on whom ruqyah is performed is a disbeliever. Ibn Baz stated that it is permissible to perform ruqyah on a disbeliever unless they are at war with Muslims. It is also permissible to instruct a disbeliever to say: "La hawla wa la quwwata illa billah (There is no might or power except with Allah)," as this falls under enjoining good.