\[ \boxed { \left \{ \left ( {t}^{2}+1 \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =-tx \left ( t \right ) +y \left ( t \right ) , \left ( {t}^{2}+1 \right ) {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =-x \left ( t \right ) -ty \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.685587 (sec), leaf count = 53 \[ \left \{\left \{x(t)\to \frac {c_1}{t^2+1}+\frac {c_2 t}{t^2+1},y(t)\to \frac {c_2}{t^2+1}-\frac {c_1 t}{t^2+1}\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 35 \[ \left \{ \left \{ x \left ( t \right ) ={\frac {{\it \_C1}\,t+{\it \_C2} }{{t}^{2}+1}},y \left ( t \right ) ={\frac {-t{\it \_C2}+{\it \_C1}}{{t} ^{2}+1}} \right \} \right \} \]