The version you found on the internet was translated from English back to French. The real quote, which is due either to La Rochefoucault or to the Abbé de Saint-Réal, is:
Nous nous tourmentons moins pour devenir heureux que pour faire croire que nous le sommes.
My translation (as literal as I could grammatically make it):
We torment ourselves less to become happy than to make others believe that we are happy.
This quote can be found in many places on the internet, most often attributed to La Rochefoucauld, but sometimes to the Abbé de Saint-Réal.
The earliest I could find it in Google books is in this 1701 book which quotes it, and attributes it to La Rochefoucauld (1638–1680).
It is in this 1705 edition of La Rochefoucauld's Maxims, where it is the sixth in a list of 50 maxims.
However, here is a link on Google books to a 1722 edition of Oeuvres de M. L'Abbé de Saint-Réal (1633-1692) which contains the exact same list of 50 maxims.
So the list of 50 maxims seems to have been incorporated into editions of La Rochefoucauld's Maxims after his death, but it wasn't in editions published when he was alive (at least, it's not in this edition that Tsundoku found, which is a reproduction of the 1668 edition, and which the editor's note claims is the last one published before La Rochefoucauld died in 1680.)
In conclusion, this maxim is by one of two 17th century writers, and was attributed to both of them in posthumous collections of their work in the 18th century. Which is the real author? I don't know whether it is possible to determine this from the resources we have. If you really want to figure this out, you might get clues by scanning Google books for the other 49 maxims in the list.