Integrate $ \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi$

Question

EDIT/UPDATE: I DO NOT NEED A SOLUTION. SEE SOS440 COMMENT FOR A FULL DETAILED SOLUTION.

Hi I am trying to integrate $$ \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi, $$ where $\alpha > |\beta|$ and min$_{0\leq \phi \leq \phi_0}(\alpha\cos \phi +\beta)\geq 0$.

I think this class of integrals is from the early 2000's mathematics journals, however I may be mistaken. Any literature on this would be very helpful as well.

Some ideas I had were trying to use trig identities to first re-write the square root expression by using $$ \frac{1}{2}(1+\cos \phi)=\cos^2\frac{\phi}{2},\quad \alpha\cos\phi+\beta=\alpha+\beta-2\alpha \sin^2\frac{\phi}{2}, $$ however I am not sure where to go from here. Thank you for reading.