The Collaborative International Dictionary of English v.0.48:

Cogitate \Cog"i*tate\, v. i. [imp. & p. p. Cogitated; p. pr. &
   vb. n. Cogitating.] [L. cogitatus, p. p. of cogitare to
   reflect upon, prob. fr. co- + the root of aio I say; hence,
   prop., to discuss with one's self. Cf. Adage.]
   To engage in continuous thought; to think.
   [1913 Webster]

         He that calleth a thing into his mind, whether by
         impression or recordation, cogitateth and considereth,
         and he that employeth the faculty of his fancy also
         cogitateth.                              --Bacon.
   [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Cogitate \Cog"i*tate\, v. t.
   To think over; to plan.
   [1913 Webster]

         He . . . is our witness, how we both day and night,
         revolving in our minds, did cogitate nothing more than
         how to satisfy the parts of a good pastor. --Foxe.
   [1913 Webster]

WordNet (r) 3.0 (2006):

cogitate
    v 1: consider carefully and deeply; reflect upon; turn over in
         one's mind
    2: use or exercise the mind or one's power of reason in order to
       make inferences, decisions, or arrive at a solution or
       judgments; "I've been thinking all day and getting nowhere"
       [syn: think, cogitate, cerebrate]

Moby Thesaurus II by Grady Ward, 1.0:

18 Moby Thesaurus words for "cogitate":
   cerebrate, conceive, conceptualize, entertain ideas, excogitate,
   exercise the mind, form ideas, give thought to, ideate,
   intellectualize, puzzle out, reason, reason out, sort out, think,
   think about, think out, think through