Basicaly, I had following equation to solve: $$\phi'(t) = \frac{a}{b\cdot\cos(\phi(t)) + c}$$
After integrating it, I get the following one: $$b\cdot\sin(\phi(t)) + c\cdot\phi(t) = a\cdot t$$ And that's where I spent hours. The only way for me is to try to guess the result, maybe I'm lack of some math techniques to solve it.
Will be glad for any advises and solutions.
This is the result. There is no way to get $\varphi(t)$ as a combination of elementary functions.