\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -x \left ( t \right ) =-3\,{t}^{2}+3\,t+1,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +y \left ( t \right ) -{t}^{2}+6\,t+1=0 \right \} } \]
Mathematica: cpu = 0.075510 (sec), leaf count = 124 \[ \left \{\left \{x(t)\to -c_2 \sin (t)+c_1 \cos (t)+\cos (t) \left (\left (3 t^2-t-13\right ) \cos (t)+(t-12) t \sin (t)\right )-\sin (t) \left (\left (-3 t^2+t+13\right ) \sin (t)+(t-12) t \cos (t)\right ),y(t)\to c_1 \sin (t)+c_2 \cos (t)+\cos (t) \left (\left (-3 t^2+t+13\right ) \sin (t)+(t-12) t \cos (t)\right )+\sin (t) \left (\left (3 t^2-t-13\right ) \cos (t)+(t-12) t \sin (t)\right )\right \}\right \} \]
Maple: cpu = 0.031 (sec), leaf count = 42 \[ \left \{ \left \{ x \left ( t \right ) =\sin \left ( t \right ) {\it \_C2} +\cos \left ( t \right ) {\it \_C1}+3\,{t}^{2}-t-13,y \left ( t \right ) = {t}^{2}-\cos \left ( t \right ) {\it \_C2}+\sin \left ( t \right ) {\it \_C1}-12\,t \right \} \right \} \]