\[ \int \frac {\cos (x)-\sin (x)}{2+\sin (2 x)} \, dx \]
Optimal antiderivative \[ \arctan \left (\cos \left (x \right )+\sin \left (x \right )\right ) \]
command
Int[(Cos[x] - Sin[x])/(2 + Sin[2*x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {1}{6} \left (3+i \sqrt {3}\right ) \arctan \left (\frac {-2 i \tan \left (\frac {x}{2}\right )-\sqrt {3}+i}{\sqrt {2 \left (1+i \sqrt {3}\right )}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \arctan \left (\frac {-2 i \tan \left (\frac {x}{2}\right )-\sqrt {3}+i}{\sqrt {2 \left (1+i \sqrt {3}\right )}}\right )-\frac {1}{6} \left (3+i \sqrt {3}\right ) \arctan \left (\frac {-2 i \tan \left (\frac {x}{2}\right )+\sqrt {3}+i}{\sqrt {2 \left (1-i \sqrt {3}\right )}}\right )-\frac {1}{6} \left (3-i \sqrt {3}\right ) \arctan \left (\frac {-2 i \tan \left (\frac {x}{2}\right )+\sqrt {3}+i}{\sqrt {2 \left (1-i \sqrt {3}\right )}}\right ) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {\cos (x)-\sin (x)}{2+\sin (2 x)} \, dx \]________________________________________________________________________________________