I can't solve this integral in Mathematica

Question

I want to calculate the following integral in Mathematica 10

$$f(x) = \frac2\pi\int_0^\infty\exp\left(-\kappa_2\frac{t^2}{2!}+\kappa_4\frac{t^4}{4!}-\dots\right)\cos\left(\kappa_1t-\kappa_3\frac{t^3}{3!}+\dots\right)\cos(xt)\mathrm dt\quad(x > 0)$$

All $\left[\kappa_i\right]$'s are given constant factors and have finite numbers.

The written codes are here:

Integrate[
  Exp[((-K[[2]]/2)*(t^2)) + ((K[[4]]/(4*3*2))*(t^4))]*
   Cos[(K[[1]]*t) - ((K[[3]]/(3*2))*(t^3))]*Cos[(x*t)], {t, 0, 
   Infinity}]/(0.5*Pi)

However, Mathematica doesn't give explicit answer. I tried it in Matlab but it doesn't work also. I am trying to solve this integral using Pade_Approximant in Mathematica but I don't know that is it possible or not? I have several questions:

1. Can I solve this integral directly? Or it needs function expansions necessarily?
2. Is Pade_Approximant best expansion method for solve this?
3. One of the questions is that exponential and cos function around what point(s) must be expanded?
4. Is Maple better software for solving this problem?

Thank you for your help

Answer 1

For specific values of $k_i$ and $x$, Mathematica has no trouble, though you will need to tell it to use numerical rather than symbolic integration (NIntegrate). For instance:

K={3.1,4.1,5.9,-2.6,5.3,5.8};
f[x_]:=NIntegrate[Exp[((-K[[2]]/2)*(t^2))+((K[[4]]/(4*3*2))*(t^4))]*Cos[(K[[1]]*t)-((K[[3]]/(3*2))*(t^3))]*Cos[(x*t)],{t,0,Infinity}]/(0.5*Pi)
Plot[f[x],{x,0,10}]

gives me

If you are looking for a symbolic solution to the cosine transform, somebody else will have to help you, as it's not something I've tried to do with Mathematica before. FourierCosTransform doesn't work with your function as-is, unfortunately.