Primitive of a continuous function over $\Bbb R$

Yes, it is possible. First compute in a standard way

Integrate[3/(5-4 Cos[x]),x]
(*2 ArcTan[3 Tan[x/2]]*)
Plot[2 ArcTan[3 Tan[x/2]],{x,-10π,10π}]

Now try the Rubi package

Get["Rubi`"]
Int[3/(5-4 Cos[x]),x]
(*x+2 ArcTan[Sin[x]/(2-Cos[x])]*)
Plot[x+2 ArcTan[Sin[x]/(2-Cos[x])],{x,-10π,10π}]

Integrate[f[x], x] gives an antiderivative whose derivative is generically equal to f[x]. From the docs:

For indefinite integrals, Integrate tries to find results that are correct for almost all values of parameters.

A primitive function of a continuous function $f$ such as the OP seeks is given by $$F(x) = \int_a^x f(t) \; dt \,.$$

prim = Integrate[3/(5 - 4 Cos[t]), {t, 0, x}, 
  Assumptions -> -Infinity < x < Infinity]

Plot[prim, {x, -30, 30}]