\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =cy \left ( t \right ) -bz \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =az \left ( t \right ) -cx \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =bx \left ( t \right ) -ay \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.065008 (sec), leaf count = 1445 \[ \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} a^2+b^2 e^{2 \sqrt {-a^2-b^2-c^2} t}+c^2 e^{2 \sqrt {-a^2-b^2-c^2} t}+b^2+c^2\right ) c_1}{2 \left (a^2+b^2+c^2\right )}-\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (a e^{\sqrt {-a^2-b^2-c^2} t} b-a b+c \sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t}+c \sqrt {-a^2-b^2-c^2}\right ) c_2}{2 \left (a^2+b^2+c^2\right )}-\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (-\sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t} b-\sqrt {-a^2-b^2-c^2} b+a c e^{\sqrt {-a^2-b^2-c^2} t}-a c\right ) c_3}{2 \left (a^2+b^2+c^2\right )},y(t)\to -\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (a e^{\sqrt {-a^2-b^2-c^2} t} b-a b-c \sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t}-c \sqrt {-a^2-b^2-c^2}\right ) c_1}{2 \left (a^2+b^2+c^2\right )}+\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (e^{2 \sqrt {-a^2-b^2-c^2} t} a^2+a^2+2 b^2 e^{\sqrt {-a^2-b^2-c^2} t}+c^2 e^{2 \sqrt {-a^2-b^2-c^2} t}+c^2\right ) c_2}{2 \left (a^2+b^2+c^2\right )}-\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (\sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t} a+\sqrt {-a^2-b^2-c^2} a+b c e^{\sqrt {-a^2-b^2-c^2} t}-b c\right ) c_3}{2 \left (a^2+b^2+c^2\right )},z(t)\to -\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (\sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t} b+\sqrt {-a^2-b^2-c^2} b+a c e^{\sqrt {-a^2-b^2-c^2} t}-a c\right ) c_1}{2 \left (a^2+b^2+c^2\right )}-\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (-\sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t} a-\sqrt {-a^2-b^2-c^2} a+b c e^{\sqrt {-a^2-b^2-c^2} t}-b c\right ) c_2}{2 \left (a^2+b^2+c^2\right )}+\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (e^{2 \sqrt {-a^2-b^2-c^2} t} a^2+a^2+2 c^2 e^{\sqrt {-a^2-b^2-c^2} t}+b^2 e^{2 \sqrt {-a^2-b^2-c^2} t}+b^2\right ) c_3}{2 \left (a^2+b^2+c^2\right )}\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 312 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}+{\it \_C2}\,\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) +{\it \_C3}\,\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) ,y \left ( t \right ) = -{\frac {1}{a \left ( {b}^{2}+{c}^{2} \right ) } \left ( \sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2 }}{\it \_C3}\,ac+\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C2}\,{a}^{2}b-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{\it \_C2}\,ac+\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C3}\,{a}^{2}b-{\it \_C1}\,{b}^{3}-{\it \_C1}\,b{c}^{2} \right ) },z \left ( t \right ) ={ \frac {1}{a \left ( {b}^{2}+{c}^{2} \right ) } \left ( \sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{ \it \_C3}\,ab-\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) { \it \_C2}\,{a}^{2}c-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{\it \_C2}\,ab-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C3}\,{a}^{2}c+{\it \_C1}\,{b}^{2}c+{\it \_C1}\,{c}^{3} \right ) } \right \} \right \} \]