Int{e^x/sqrt(1+e^x)}dx

substitution& sqrt(e^x+1)=t& =>& e^x=t^2-1& =>& x=ln(t^2-1)& =>& dx=2tdt/(t^2-1)
limits for new variable: x=0 => t=sqrt(2)
x=ln3 => t=2
so we have
Int{(t^2-1)/t*2t/(t^2-1)}dt=Int{2}dt=t
Using Newton-Leibniz formula for limits sqrt(2),2 we obtain
2(2-sqrt(2))=4-2sqrt(2) and that's it