The Oxford dictionary has ataraxia (ataraxy) as a valid word but not ataraxis. however, I've seen and heard the ataraxis being used once in while. But it happens that the guys at Oxford do not recognize it (ataraxis) as a word. You can search ataraxis and many sites, along with Google itself, will define it. Now the question is whether I can use the word ataraxis and ataraxy (ataraxia) in place of each other and will any examiner such as those at GRE and GMAT consider and approve it.
Who would be a father to go through such an ordeal ? — Thank heaven I am still a bachelor ! Give me the Pyrrhonic beatitudes of ataraxis, — the pococurante tranquillity of a luxurious indifference, — before all the family sensibilities in the world !
-Cecil, a peer : a sequel to Cecil, or, The adventures ... v.2. Gore, Mrs. (Catherine Grace Frances), 1799-1861.
A mention of the coinage that was published in 1930:
Karl Barth, a master of the art of expression and gifted with an exu- berant imagination, revels in lengthy sentences (see one of nearly three hundred words on page 168) striking compounds, new coin- ages (Da-Sein So-Sein, Aseitaet, Ataraxis, Amphibolie, Inkonzin- nitaet, etc.) and involved constructions, cryptic, if not cabalistic.1 Like the university style of German literature, Barth's diction is exceptionally complex and not infrequently requires, on account of a superior inflectional system in German, a breaking up of a sentence into two in order to reach tolerable sense in English.
-The Karl Barth theology; or The new transcendentalism. ... . Zerbe, Alvin Sylvester, 1847-1935.
Incidentally, I used the word myself last night -how odd
Ataraxia is a central term in Epicurean philosophy, and this is the way it is normally spelled.
Greek tarattô means "to perturb", and a-tarax-ia is a regularly formed derivation meaning a state of "unperturbedness", mainly of the mind, when one manages to remain unworried. It is similar to the Stoic ideal of apatheia "unfeelingness", which is stronger and more rigid, being unmoved by any emotion at all.
Most Greek and Latin words on -ia become -y in English. But for some reason the Latin spelling ataraxia is the one in common use in modern academic literature; I would not recommend using ataraxy in an exam.
Ataraxis would be a perfectly valid derivation in Greek, "[the action of] being unperturbed", with a similar meaning. But for some reason this is not the form Epicurus and Lucretius chose, as far as I know, and it never came into common use.
Google Ngrams bears this out: