Dear Jofre,
are you sure that your syntax is ok? In your expression
Integrate[Cos x/(R^2 + r^2 - (2*R*r*Cos x) + (k - k1)^2)^1/2, {x, 0, 2 \[Pi]}]
the Cos has no proper argument - there are no square brackets. And there is a difference to that and Cos[x]:

If I use
Integrate[Cos[x]/(R^2 + r^2 - (2*R*r*Cos[x]) + (k - k1)^2)^1/2, {x, 0, 2 \[Pi]}]
the result is somewhat different.

I suppose that the original question is about WolframAlpha, where none of this matters, because it does not evaluate.
As a suggestion to Ali Kreem I would use the free tier of the WolframCloud. It allows you to use proper Mathematica syntax and your integral will evaluate.
Cheers,
Marco