Evaluating $\int\frac{1}{1+3\cos^2 x}dx$

Question

I got this fairly simple looking problem $$\int\frac{1}{1+3\cos^2 x}dx$$ Although it looks simple enough but I am stuck on how to begin a hint might be enough to solve this problem. What I tried was that changing $\cos^2 x$ to $\sin^2 x$ $$\int\frac{1}{1+3(1-\sin^2 x)}dx$$ I don't know what else to do. Any formula that I am missing here? I did try some half angle formula for $\cos x$ but it was making things bit messy because of square, simplest hint must be enough for this.

Answer 1

This is directly from Wolfram Alpha.

Answer 2

Here is an easy way: Perform the Weierstrass substitution on the cosine term, comes out beautifully.