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Answering to Definite Integral post, I tried to do the first example in ?inttrans[fourier], 
assume(a>0):
inttrans[fourier](3/(a^2 + t^2),t,w);

         Pi (exp(a~ w) Heaviside(-w) + exp(-a~ w) Heaviside(w))
       3 ------------------------------------------------------
                                   a~
using int, 
int(3/(a^2+t^2)*exp(-I*w*t),t=-infinity..infinity);

                                  0
No comment. Well, I tried another example from the same help page, 
inttrans[fourier](1/(4 - I*t)^(1/3),t,2+w);

             1/2
            3    GAMMA(2/3) exp(-8 - 4 w) Heaviside(2 + w)
            ----------------------------------------------
                                     2/3
                              (2 + w)

int(1/(4 - I*t)^(1/3)*exp(-I*t*w),t=-infinity..infinity);

                               /
           1/3  1/2            |         1/2              1/3
  - 1/4 I 2    3    GAMMA(2/3) |1/10 Pi 3    (-1 + 12 w) 2
                               |
                               \

                                              /              1/3
        exp(-2 w) WhittakerM(2/3, 5/6, 4 w)  /  (GAMMA(2/3) w   )
                                            /

                   1/2  1/3
               Pi 3    2    exp(-2 w) WhittakerM(5/3, 5/6, 4 w)
         + 3/5 ------------------------------------------------
                                           1/3
                               GAMMA(2/3) w

                                                            \
                   1/2      (1/3)  (2/3)     (1/6)          |
         + 2/3 Pi 3    (w I)      2      (-1)      exp(-4 w)|/(Pi w)
                                                            |
                                                            /

                                      /
                  1/3  1/2            |         1/2              1/3
         + 1/4 I 2    3    GAMMA(2/3) |1/10 Pi 3    (-1 + 12 w) 2
                                      |
                                      \

                                              /              1/3
        exp(-2 w) WhittakerM(2/3, 5/6, 4 w)  /  (GAMMA(2/3) w   )
                                            /

                   1/2  1/3
               Pi 3    2    exp(-2 w) WhittakerM(5/3, 5/6, 4 w)
         + 3/5 ------------------------------------------------
                                           1/3
                               GAMMA(2/3) w

                                                             \
                   1/2       (1/3)  (2/3)     (5/6)          |
         - 2/3 Pi 3    (-I w)      2      (-1)      exp(-4 w)|/(Pi w)
                                                             |
                                                             /

evalf(eval(%,w=1))=evalf(eval(%%,w=1));

                                                     -5
               0.04295751906 + 0. I = 0.6927914186 10
Again I'm silent. OK, another integral from the same page, 
inttrans[fourier](BesselJ(0,4*(t^2 + 1)^(1/2)), t, s); 
                        2                                          /
  8 exp(s I) cos(-16 + s ) (Heaviside(s + 4) - Heaviside(s - 4))  /
                                                                 /

               2 1/2
        (16 - s )

int(BesselJ(0,4*(t^2 + 1)^(1/2))*exp(-I*t*w),t=-infinity..infinity);

            infinity
           /
          |                         2     1/2
          |          BesselJ(0, 4 (t  + 1)   ) exp(-I w t) dt
          |
         /
           -infinity
That's better - the integral is unevaluated, but at least the answer is not wrong.
integral fourier inttrans

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