An example using SymPy.stats to compute the characteristic function of a general Cauchy random variable. This is run in isympy (hence the pretty printing).

cauchy_characteristic

In [1]: from sympy.stats import *

In [2]: x0 = Symbol('x0', real=True, bounded=True)

In [3]: gamma = Symbol('gamma', positive=True)

In [4]: X = Cauchy('X', x0, gamma)

In [5]: density(X)(x)
Out[5]: 
         1         
───────────────────
    ⎛            2⎞
    ⎜    (x - x₀) ⎟
π⋅γ⋅⎜1 + ─────────⎟
    ⎜         2   ⎟
    ⎝        γ    ⎠

In [6]: t = Symbol('t', positive=True)

In [7]: E(exp(I*t*X))
Out[7]: 
  ⎛  ⎛              π⎞                 ⎛           ⅈ⋅π⎞          ⎞  ⅈ⋅t⋅x₀   ⎛  ⎛             π⎞                 ⎛           ⅈ⋅π⎞          ⎞  ⅈ⋅t⋅x₀
  ⎜2⋅⎜-ⅈ⋅Shi(γ⋅t) - ─⎟⋅cosh(γ⋅t) + 2⋅ⅈ⋅⎜Chi(γ⋅t) - ───⎟⋅sinh(γ⋅t)⎟⋅ℯ         ⎜2⋅⎜ⅈ⋅Shi(γ⋅t) - ─⎟⋅cosh(γ⋅t) - 2⋅ⅈ⋅⎜Chi(γ⋅t) + ───⎟⋅sinh(γ⋅t)⎟⋅ℯ      
  ⎝  ⎝              2⎠                 ⎝            2 ⎠          ⎠           ⎝  ⎝             2⎠                 ⎝            2 ⎠          ⎠        
- ──────────────────────────────────────────────────────────────────────── - ───────────────────────────────────────────────────────────────────────
                                    2⋅π                                                                        2⋅π                                  

In [8]: simplify(E(exp(I*t*X)))
Out[8]: 
                          ⅈ⋅t⋅x₀
(-sinh(γ⋅t) + cosh(γ⋅t))⋅ℯ      

What sympy version do you use? I am trying the same thing but I got

In [9]: simplify(E(exp(I*t*X)))
Out[9]: 
  ∞                           
  ⌠                           
  ⎮           ⅈ⋅X⋅t           
  ⎮          ℯ                
γ⋅⎮  ────────────────────── dX
  ⎮   2             2     2   
  ⎮  X  - 2⋅X⋅x₀ + γ  + x₀    
  ⌡                           
  -∞                          
──────────────────────────────
              π      

which is not as nice as yours.