\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =y \left ( t \right ) -z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =x \left ( t \right ) +y \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =x \left ( t \right ) +z \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.010001 (sec), leaf count = 105 \[ \left \{\left \{x(t)\to c_2 \left (e^t-1\right )+c_3 \left (1-e^t\right )+c_1,y(t)\to c_1 \left (e^t-1\right )+c_2 \left (e^t t+1\right )+c_3 \left (-e^t t+e^t-1\right ),z(t)\to c_1 \left (e^t-1\right )+c_2 \left (e^t t-e^t+1\right )+c_3 \left (-e^t t+2 e^t-1\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 48 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C2}+{\it \_C3}\,{{\rm e}^ {t}},y \left ( t \right ) ={\it \_C3}\,{{\rm e}^{t}}t+{\it \_C1}\,{ {\rm e}^{t}}-{\it \_C2},z \left ( t \right ) ={\it \_C3}\,{{\rm e}^{t}}t +{\it \_C1}\,{{\rm e}^{t}}-{\it \_C3}\,{{\rm e}^{t}}-{\it \_C2} \right \} \right \} \]