Ok this is what I came up with:
First I change $\dot x$ =$\cos(x+t)$ with the change of variables $u=x+t$ then $\dot u=\dot x+1$ $\Rightarrow$ $\dot u=cos(u)+1$, then I solve by separation by parts which I can do if $u\in(-\pi,\pi)$.
Then $\frac{\dot u}{\cos(u) +1}=1 \Rightarrow \int\frac{\dot u}{\cos(u) +1}dt=t+C_1 \Rightarrow \tan(\frac{u}{2})+C_2 =t+C_1 \Rightarrow \tan(\frac{u}{2})=t+C_3$ where $C_3=C_1-C_2$
But then $u=2\arctan(t+C_3)$, by undoing the change of variables $u=x+t \Rightarrow x=2\arctan(t+C_3)-t$
Am I right to say x doesnt have a maximum and minima since if $T$ is large enough then $x\approx-(t+\pi)$ as that function diverges in the future and the past? What happened to the restriction of u shouldn't it have given me some bound on x?
Then for the inflection points i'm supposed to derivate twice that function or is there a more qualitative way of calculating them?