\[ \left \{x'(t)=x(t)+y(t)-z(t),y'(t)=-x(t)+y(t)+z(t),z'(t)=x(t)-y(t)+z(t)\right \} \] ✓ Mathematica : cpu = 0.0555281 (sec), leaf count = 177
\[\left \{\left \{x(t)\to \frac {1}{3} e^t \left (\sqrt {3} \left (c_2-c_3\right ) \sin \left (\sqrt {3} t\right )+\left (2 c_1-c_2-c_3\right ) \cos \left (\sqrt {3} t\right )+c_1+c_2+c_3\right ),y(t)\to \frac {1}{3} e^t \left (-\sqrt {3} \left (c_1-c_3\right ) \sin \left (\sqrt {3} t\right )-\left (c_1-2 c_2+c_3\right ) \cos \left (\sqrt {3} t\right )+c_1+c_2+c_3\right ),z(t)\to \frac {1}{3} e^t \left (\sqrt {3} \left (c_1-c_2\right ) \sin \left (\sqrt {3} t\right )-\left (c_1+c_2-2 c_3\right ) \cos \left (\sqrt {3} t\right )+c_1+c_2+c_3\right )\right \}\right \}\]
✓ Maple : cpu = 0.082 (sec), leaf count = 120
\[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{t}} \left ( \sin \left ( \sqrt {3}t \right ) {\it \_C2}+\cos \left ( \sqrt {3}t \right ) {\it \_C3}+{\it \_C1} \right ) ,y \left ( t \right ) ={\frac {{{\rm e}^{t}} \left ( {\it \_C2}\,\sqrt {3}-{\it \_C3} \right ) \cos \left ( \sqrt {3}t \right ) }{2}}+{\frac {{{\rm e}^{t}} \left ( -{\it \_C3}\,\sqrt {3}-{\it \_C2} \right ) \sin \left ( \sqrt {3}t \right ) }{2}}+{\it \_C1}\,{{\rm e}^{t}},z \left ( t \right ) ={\frac {{{\rm e}^{t}} \left ( -{\it \_C2}\,\sqrt {3}-{\it \_C3} \right ) \cos \left ( \sqrt {3}t \right ) }{2}}+{\frac {{{\rm e}^{t}} \left ( {\it \_C3}\,\sqrt {3}-{\it \_C2} \right ) \sin \left ( \sqrt {3}t \right ) }{2}}+{\it \_C1}\,{{\rm e}^{t}} \right \} \right \} \]