\[ \left \{x'(t)=x(t)+y(t)-z(t),y'(t)=-x(t)+y(t)+z(t),z'(t)=x(t)-y(t)+z(t)\right \} \] ✓ Mathematica : cpu = 0.0299694 (sec), leaf count = 278
DSolve[{Derivative[1][x][t] == x[t] + y[t] - z[t], Derivative[1][y][t] == -x[t] + y[t] + z[t], Derivative[1][z][t] == x[t] - y[t] + z[t]},{x[t], y[t], z[t]},t]
\[\left \{\left \{x(t)\to \frac {1}{3} c_1 e^t \left (2 \cos \left (\sqrt {3} t\right )+1\right )-\frac {1}{3} c_2 e^t \left (-\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right )-\frac {1}{3} c_3 e^t \left (\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right ),y(t)\to \frac {1}{3} c_2 e^t \left (2 \cos \left (\sqrt {3} t\right )+1\right )-\frac {1}{3} c_3 e^t \left (-\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right )-\frac {1}{3} c_1 e^t \left (\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right ),z(t)\to \frac {1}{3} c_3 e^t \left (2 \cos \left (\sqrt {3} t\right )+1\right )-\frac {1}{3} c_1 e^t \left (-\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right )-\frac {1}{3} c_2 e^t \left (\sqrt {3} \sin \left (\sqrt {3} t\right )+\cos \left (\sqrt {3} t\right )-1\right )\right \}\right \}\] ✓ Maple : cpu = 0.059 (sec), leaf count = 110
dsolve({diff(x(t),t) = x(t)+y(t)-z(t), diff(y(t),t) = y(t)+z(t)-x(t), diff(z(t),t) = z(t)+x(t)-y(t)})
\[\left \{x \left (t \right ) = {\mathrm e}^{t} \left (c_{1}+\sin \left (\sqrt {3}\, t \right ) c_{2}+\cos \left (\sqrt {3}\, t \right ) c_{3}\right ), y \left (t \right ) = {\mathrm e}^{t} \left (\left (\frac {\sqrt {3}\, c_{2}}{2}-\frac {c_{3}}{2}\right ) \cos \left (\sqrt {3}\, t \right )+\left (-\frac {\sqrt {3}\, c_{3}}{2}-\frac {c_{2}}{2}\right ) \sin \left (\sqrt {3}\, t \right )+c_{1}\right ), z \left (t \right ) = \left (\left (-\frac {\sqrt {3}\, c_{2}}{2}-\frac {c_{3}}{2}\right ) \cos \left (\sqrt {3}\, t \right )+\left (\frac {\sqrt {3}\, c_{3}}{2}-\frac {c_{2}}{2}\right ) \sin \left (\sqrt {3}\, t \right )+c_{1}\right ) {\mathrm e}^{t}\right \}\]