Well...FWIW:

```
sage: var("t, omega")
(t, omega)
sage: from sympy.integrals.transforms import fourier_transform as sft
sage: from sympy.integrals.transforms import inverse_fourier_transform as isft
sage: sft?
Signature: sft(f, x, k, **hints)
Docstring:
Compute the unitary, ordinary-frequency Fourier transform of f,
defined as
F(k) = int_{-infty}^infty f(x) e^{-2pi i x k} d x. Signature: sft(f, x, k, **hints)
Docstring:
Compute the unitary, ordinary-frequency Fourier transform of f,
defined as
F(k) = int_{-infty}^infty f(x) e^{-2pi i x k} d x.
If the transform cannot be computed in closed form, this function
returns an unevaluated "FourierTransform" object.
For other Fourier transform conventions, see the function
"sympy.integrals.transforms._fourier_transform()".
For a description of possible hints, refer to the docstring of
"sympy.integrals.transforms.IntegralTransform.doit()". Note that
for this transform, by default "noconds=True".
If the transform cannot be computed in closed form, this function
returns an unevaluated "FourierTransform" object.
For other Fourier transform conventions, see the function
"sympy.integrals.transforms._fourier_transform()".
For a description of possible hints, refer to the docstring of
"sympy.integrals.transforms.IntegralTransform.doit()". Note that
for this transform, by default "noconds=True".
```

[ Abriged...]

```
sage: sft(e^(-t^2)*(1-t^2), t, omega)._sage_()
1/2*sqrt(pi)*(2*pi^2*omega^2 + 1)*e^(-pi^2*omega^2)
sage: isft(sft(e^(-t^2)*(1-t^2), t, omega)._sage_(), omega, t)._sage_()
-(t^2 - 1)*e^(-t^2)
```

**EDITED :** The case you propose seems hard to integrate: `sympy`

doesn't return a result for your second case. BTW, neither `maxima`

, `sympy`

, `fricas`

nor `giac`

are able to compute the integral directly. Neither does `mathematica`

, which seems to be able to compute the Fourier integral by less direct means :

```
sage: mathematica.Integrate(g(t)*e^(-2*I*pi*t*o),[t,-oo,oo])
Integrate[(E^((-2*I)*o*Pi*t - t^2)*Sin[t])/t, {t, -Infinity, Infinity}]
sage: mathematica.FourierTransform(e^(-t^2)*sin(t)/t, t, omega).sage()
1/2*sqrt(1/2)*sqrt(pi)*erf(1/2*omega + 1/2) + 1/2*sqrt(1/2)*sqrt(pi)*erf(-1/2*omega + 1/2)
```

**WARNING** All the engines do not follow the same conventins for Fourier transform parameters. See Wikipedia for alternatives...

HTH,