How to solve this integral manually

Question

I was very much surprised that the Wolfram Online Integrator solved this integral very readably and in an elegant way :

$${\large\int}\frac{\cos\left(\left(11+\frac 12\right)x\right)}{\cos\left(\left(11-\frac 12\right)x\right)\sin(x)}\,{\rm d}x$$ So what I am looking for now is the step-by-step solution of $${\large\int}\frac{\cos\left(\left(m+\frac 12\right)x\right)}{\cos\left(\left(m-\frac 12\right)x\right)\sin(x)}\,{\rm d}x$$ for m integer which should lead to the ( obvious ) generalization of the formula CAS-calculated for the case e.g. m=11.

I think I have not enough depth of knowledge in the Chebyshev polynomials ( of all 4 kinds ) to solve this problem.

Answer 1

HINT:

$$\cos\left(\left(11+\frac12\right)x\right)=\cos\left(\underbrace{\left(11-\frac12\right)x}+x\right)$$

Apply $\cos(A+B)=\cos A\cos B-\sin A\sin B$

Answer 2

$\cos\left(\left(m+\frac{1}{2}\right)x\right)=\cos\left(\left(m-\frac{1}{2}\right)x\right)\cos x-\sin\left(\left(m-\frac{1}{2}\right)x\right)\sin x$