3 Mathematica. Time used: 0.671 (sec). Leaf size: 19

ode=D[y[x],x]==1+1/x-1/(y[x]^2+2)-1/(x*(y[x]^2+2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]

\begin{align*} y(x)&\to \text {InverseFunction}[\arctan (\text {$\#$1})+\text {$\#$1}\&][x+\log (x)+c_1] \end{align*}