Mathematica Integrate just spits out the input ... Integral does not solve

Question

Just find some math help. When I try to calculate this integral:

Integrate[Cos[t]/(1 + b^2 t^2 - (2*b*c*t)/a + c^2/a^2)^(3/2), { t, -Infinity, Infinity}, Assumptions -> {a, b, c} \[Element] Reals] 

Math just spits out:

 Integrate[Cos[t]/(1 + c^2/a^2 - (2 bct)/a + b^2 t^2)^(3/2), {t, -\[Infinity], \[Infinity]}, Assumptions -> (a | b | c) \[Element] Reals] 

How would I rate this integral?

Answer 1

Simplify. First of all, you do not need 3 independent parameters, just two. Then this integral is equivalent (by changing variables and re-scaling the parameters)

 Integrate[Cos[(a - bt)]/(1 + t^2)^(3/2), {t, -Infinity, Infinity}, Assumptions -> {a, b} \[Element] Reals] 

with the answer:

2 Abs [b] BesselK [1, Abs [b]] Cos [a]

The constants a, b are different from the original ones, but can be expressed through them if you perform simple scaling and changing the variable. And here is your beautiful function in the parameter space:

 Plot3D[2 Abs[b] BesselK[1, Abs[b]] Cos[a], {a, -5, 5}, {b, -5, 5}, PlotRange -> All, Mesh -> All, ColorFunction -> "DarkRainbow", MeshStyle -> Opacity[.1], AxesLabel -> {a, b}]