\[ \left \{x'(t)=c y(t)-b z(t),y'(t)=a z(t)-c x(t),z'(t)=b x(t)-a y(t)\right \} \] ✓ Mathematica : cpu = 0.0698042 (sec), leaf count = 1084
\[\left \{\left \{x(t)\to \frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (2 e^{\sqrt {-a^2-b^2-c^2} t} c_1 a^2+b^2 \left (1+e^{2 \sqrt {-a^2-b^2-c^2} t}\right ) c_1+c^2 \left (1+e^{2 \sqrt {-a^2-b^2-c^2} t}\right ) c_1-c \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (\sqrt {-a^2-b^2-c^2} \left (1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_2+a \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_3\right )-b \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (a \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_2-\sqrt {-a^2-b^2-c^2} \left (1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_3\right )\right )}{2 \left (a^2+b^2+c^2\right )},y(t)\to \frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (\left (1+e^{2 \sqrt {-a^2-b^2-c^2} t}\right ) c_2 a^2-\left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (b \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_1+\sqrt {-a^2-b^2-c^2} \left (1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_3\right ) a+2 b^2 e^{\sqrt {-a^2-b^2-c^2} t} c_2+c^2 \left (1+e^{2 \sqrt {-a^2-b^2-c^2} t}\right ) c_2+c \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (\sqrt {-a^2-b^2-c^2} \left (1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_1-b \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_3\right )\right )}{2 \left (a^2+b^2+c^2\right )},z(t)\to \frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (\left (1+e^{2 \sqrt {-a^2-b^2-c^2} t}\right ) c_3 a^2-\left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (c \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_1-\sqrt {-a^2-b^2-c^2} \left (1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_2\right ) a-b \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (\sqrt {-a^2-b^2-c^2} \left (1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_1+c \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) c_2\right )+2 c^2 e^{\sqrt {-a^2-b^2-c^2} t} c_3+b^2 \left (1+e^{2 \sqrt {-a^2-b^2-c^2} t}\right ) c_3\right )}{2 \left (a^2+b^2+c^2\right )}\right \}\right \}\]
✓ Maple : cpu = 0.085 (sec), leaf count = 257
\[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}+{\it \_C2}\,\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) +{\it \_C3}\,\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) ,y \left ( t \right ) ={\frac {1}{a \left ( {b}^{2}+{c}^{2} \right ) } \left ( \left ( -{a}^{2}b{\it \_C3}+ac{\it \_C2}\,\sqrt {{a}^{2}+{b}^{2}+{c}^{2}} \right ) \cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) + \left ( -{a}^{2}b{\it \_C2}-ac{\it \_C3}\,\sqrt {{a}^{2}+{b}^{2}+{c}^{2}} \right ) \sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) +b{\it \_C1}\, \left ( {b}^{2}+{c}^{2} \right ) \right ) },z \left ( t \right ) ={\frac {1}{a \left ( {b}^{2}+{c}^{2} \right ) } \left ( \left ( -{a}^{2}c{\it \_C3}-ab{\it \_C2}\,\sqrt {{a}^{2}+{b}^{2}+{c}^{2}} \right ) \cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) + \left ( -{a}^{2}c{\it \_C2}+ab{\it \_C3}\,\sqrt {{a}^{2}+{b}^{2}+{c}^{2}} \right ) \sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) +c{\it \_C1}\, \left ( {b}^{2}+{c}^{2} \right ) \right ) } \right \} \right \} \]