2.2.5.3 ✓ Maple. Time used: 0.007 (sec). Leaf size: 18

ode:=diff(y(x),x) = 1+1/x-1/(y(x)^2+2)-1/x/(y(x)^2+2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y = \tan \left (\operatorname {RootOf}\left (x +\ln \left (x \right )-\tan \left (\textit {\_Z} \right )-\textit {\_Z} +c_1 \right )\right ) \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=1+\frac {1}{x}-\frac {1}{y \left (x \right )^{2}+2}-\frac {1}{x \left (y \left (x \right )^{2}+2\right )} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=1+\frac {1}{x}-\frac {1}{y \left (x \right )^{2}+2}-\frac {1}{x \left (y \left (x \right )^{2}+2\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\left (\frac {d}{d x}y \left (x \right )\right ) \left (y \left (x \right )^{2}+2\right )}{y \left (x \right )^{2}+1}=\frac {x +1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\left (\frac {d}{d x}y \left (x \right )\right ) \left (y \left (x \right )^{2}+2\right )}{y \left (x \right )^{2}+1}d x =\int \frac {x +1}{x}d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )+\arctan \left (y \left (x \right )\right )=x +\ln \left (x \right )+\mathit {C1} \end {array} \]