I am trying to find closed forms for integrals of rational functions times a trigonometric function.

I obtained some totally astonishing results like

\(\displaystyle \int_0^\infty \frac{\cos x}{5-4\cos x} \times \frac{1}{1+x^2}dx = \frac{\pi }{2e-1}\left[ \frac{e^2-1}{e-2}\left\{ \frac{2}{3}-\frac{e}{e^2-1}\right\}-\frac{e}{2} \right]=0.5568468705 \cdots\)

Mathematica can't evaluate this integral.

Most of my results are recorded here.