$$2\int_a^b \tan^{-1} \left(\frac{a \sin x}{b+a\cos x}\right) \, dx \geq (b-a) \left( {\sqrt{ab}}+\sqrt{(a^2+b^2)/2} \, \right)$$
We have to show this. I have only written a part of the complete problem. If I am able to solve this, I can tackle the rest. The actual problem is in cyclic summation. One condition : $0 < a < b< \pi/2$