$\cfrac{{2 + \sin 2x}}{{1 + \cos 2x}}{e^x}$

Let $I = \int {\cfrac{{{e^x}\left( {2 + \sin 2x} \right)}}{{1 + \cos 2x}}dx} = \int {{e^x}\left( {\cfrac{{2 + 2\sin x\cos x}}{{2{{\cos }^2}x}}} \right)} dx$

$= \int {{e^x}\left[ {{{\sec }^2}x + \tan x} \right]dx} = \int {{e^x}\left( {\tan x + {{\sec }^2}x} \right)dx}$

$= \int {{e^x}\left[ {\tan x + \cfrac{d}{{dx}}\left( {\tan x} \right)} \right]dx} = {e^x}\tan x + C$