\[ \left \{x'(t)=x(t)+y(t)-z(t),y'(t)=-x(t)+y(t)+z(t),z'(t)=x(t)-y(t)+z(t)\right \} \] ✓ Mathematica : cpu = 0.0501751 (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 = 0.073 (sec), leaf count = 128
\[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{t}} \left ( \sin \left ( \sqrt {3}t \right ) {\it \_C2}+\cos \left ( \sqrt {3}t \right ) {\it \_C3}+{\it \_C1} \right ) ,y \left ( t \right ) =-{\frac {{{\rm e}^{t}} \left ( \sqrt {3}\sin \left ( \sqrt {3}t \right ) {\it \_C3}-\sqrt {3}\cos \left ( \sqrt {3}t \right ) {\it \_C2}+\sin \left ( \sqrt {3}t \right ) {\it \_C2}+\cos \left ( \sqrt {3}t \right ) {\it \_C3}-2\,{\it \_C1} \right ) }{2}},z \left ( t \right ) ={\frac {{{\rm e}^{t}} \left ( \sqrt {3}\sin \left ( \sqrt {3}t \right ) {\it \_C3}-\sqrt {3}\cos \left ( \sqrt {3}t \right ) {\it \_C2}-\sin \left ( \sqrt {3}t \right ) {\it \_C2}-\cos \left ( \sqrt {3}t \right ) {\it \_C3}+2\,{\it \_C1} \right ) }{2}} \right \} \right \} \]