Use a double-angle identity and an exponential substitution

Use the source double-angle guidance: $1+\cos2x=2\cos^2x$. Therefore the integral is $\frac12\int\sec^2x\,dx=\frac12\tan x+C$.

With $u=e^x+2$, $du=e^x dx=(u-2)dx$. The integral becomes $\int\dfrac{(u-2)^2}{u}\dfrac{du}{u-2}=\int(1-2/u)\,du=u-2\ln|u|+C$. Substitute back: $e^x+2-2\ln|e^x+2|+C$. The absolute-value bars match the key; the argument is actually positive.