\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =x \left ( t \right ) +y \left ( t \right ) -z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =y \left ( t \right ) +z \left ( t \right ) -x \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =z \left ( t \right ) +x \left ( t \right ) -y \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.052007 (sec), leaf count = 278 \[ \left \{\left \{x(t)\to \frac {1}{3} c_1 e^t \left (2 \cos \left (\sqrt {3} t\right )+1\right )-\frac {1}{3} c_2 e^t \left (-\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right )-\frac {1}{3} c_3 e^t \left (\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right ),y(t)\to \frac {1}{3} c_2 e^t \left (2 \cos \left (\sqrt {3} t\right )+1\right )-\frac {1}{3} c_3 e^t \left (-\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right )-\frac {1}{3} c_1 e^t \left (\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right ),z(t)\to \frac {1}{3} c_3 e^t \left (2 \cos \left (\sqrt {3} t\right )+1\right )-\frac {1}{3} c_1 e^t \left (-\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right )-\frac {1}{3} c_2 e^t \left (\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right )\right \}\right \} \]
Maple: cpu = 0.063 (sec), leaf count = 128 \[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{t}} \left ( \sin \left ( \sqrt {3}t \right ) {\it \_C2}+\cos \left ( \sqrt {3}t \right ) { \it \_C3}+{\it \_C1} \right ) ,y \left ( t \right ) =-{\frac {{{\rm e}^{t }} \left ( \sin \left ( \sqrt {3}t \right ) \sqrt {3}{\it \_C3}-\cos \left ( \sqrt {3}t \right ) \sqrt {3}{\it \_C2}+\sin \left ( \sqrt {3}t \right ) {\it \_C2}+\cos \left ( \sqrt {3}t \right ) {\it \_C3}-2\,{\it \_C1} \right ) }{2}},z \left ( t \right ) ={\frac {{{\rm e}^{t}} \left ( \sin \left ( \sqrt {3}t \right ) \sqrt {3}{\it \_C3}-\cos \left ( \sqrt { 3}t \right ) \sqrt {3}{\it \_C2}-\sin \left ( \sqrt {3}t \right ) {\it \_C2}-\cos \left ( \sqrt {3}t \right ) {\it \_C3}+2\,{\it \_C1} \right ) }{2}} \right \} \right \} \]