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Mathematica can't calculate Fourier transform (Dirac mean position eigenfunction)

Hi, I'm attempting to use Mathematica to calculate a mean-position
eigenfunction of the Dirac equation. To do so I need to evaluate
Fourier transforms (from k-space to r-space) of wavefunctions dependent

( (1 + k^2 + (1 + k^2)^(1/2) )^(-1/2)

where k is in units of the Compton wavevector.

Cell expression:

        RowBox[{"1", "+",
          SuperscriptBox["k", "2"], "+",
            RowBox[{"1", "+",
              SuperscriptBox["k", "2"]}]]}]]]], "Output"]

Mathematica is unable to evaluate the FT of the above (either Fourier
sine transform or normal FT). Can anyone give any suggestions as to how
I could evaluate it?

More specifically, I am making a reverse Foldy-Wouthuysen
transformation of a mean-position eigenfunction in p-space, then
transforming the result into r-space assuming spherical symmetry. The
first component of the r-space eigenfunction is given by the Fourier
sine transform of:

k ( 1 + (1 + k^2)^(-1/2) )^(1/2)

    RowBox[{"k", " ",
        RowBox[{"1", "+",
              RowBox[{"1", "+",
                SuperscriptBox["k", "2"]}]]]}]]}]], "Output"]

Thanks for any help.

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