I have tried to find antiderivative of
$$ \frac{\cos x\ }{2+\sin 2x} $$
using the variable change $t= \cos x -\sin x$ with sin $\sin2x=2\sin x\cos x $. But i don't come up to its closed-form result as shown below.

How can I find its antiderivative? Thanks in advance

[![enter image description here](https://i.stack.imgur.com/i2CKh.gif)](https://i.stack.imgur.com/i2CKh.gif)