3 Mathematica. Time used: 2.969 (sec). Leaf size: 59

ode=(1+v[u]^2)+(1+u^2)*v[u]*D[v[u],u]==0; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]

\begin{align*} v(u)&\to -\sqrt {-1+e^{-2 \arctan (u)+2 c_1}}\\ v(u)&\to \sqrt {-1+e^{-2 \arctan (u)+2 c_1}}\\ v(u)&\to -i\\ v(u)&\to i \end{align*}