Integrate[(E^(I*k0*x)*(-1 + E^(-(c*x)))^2*BesselJ[1, k*x])/(k0^3*x^2), {x, 0, Infinity}, Assumptions -> n == 1 && q == 3 && κ == 2]

                                -2 c x + I k0 x        c x 2                                    2  2              4  4              6  6               8  8      Pi                                         2  2               4  4              6  6               8  8      Pi
                             -(E                (-1 + E   )  (8 k x (-14783093325 + 1452971520 k  x  - 309657600 k  x  + 251658240 k  x  + 2147483648 k  x ) Cos[-- + k x] - 3 (156043762875 - 11416204800 k  x  + 1589575680 k  x  - 587202560 k  x  + 2147483648 k  x ) Sin[-- + k x]))
                                                                                                                                                                 4                                                                                                            4
Integrate::idiv: Integral of ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ does not converge on {0, Infinity}.
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                                                                                                                                        8589934592 k     k0  Sqrt[2 Pi] x
Series[Integrate[(E^(I*k0*x)*(-1 + E^(-(c*x)))^2*BesselJ[1, k*x])/(k0^3*x^2), {x, 0, Infinity}, Assumptions -> n == 1 && q == 3 && κ == 2], {k, Infinity, 10}]