\[ \boxed { \left \{ a{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =bc \left ( y \left ( t \right ) -z \left ( t \right ) \right ) ,b{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =ca \left ( z \left ( t \right ) -x \left ( t \right ) \right ) ,c{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =ab \left ( x \left ( t \right ) -y \left ( t \right ) \right ) \right \} } \]
Mathematica: cpu = 0.090011 (sec), leaf count = 1304 \[ \left \{\left \{x(t)\to \frac {e^{-i \sqrt {a^2+b^2+c^2} t} \left (2 e^{i \sqrt {a^2+b^2+c^2} t} a^2+b^2 e^{2 i \sqrt {a^2+b^2+c^2} t}+c^2 e^{2 i \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 {b e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (a e^{i \sqrt {a^2+b^2+c^2} t} b-a b+i c \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t}+i c \sqrt {a^2+b^2+c^2}\right ) c_2}{2 a \left (a^2+b^2+c^2\right )}-\frac {c e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (-i \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t} b-i \sqrt {a^2+b^2+c^2} b+a c e^{i \sqrt {a^2+b^2+c^2} t}-a c\right ) c_3}{2 a \left (a^2+b^2+c^2\right )},y(t)\to -\frac {a e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (a e^{i \sqrt {a^2+b^2+c^2} t} b-a b-i c \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t}-i c \sqrt {a^2+b^2+c^2}\right ) c_1}{2 b \left (a^2+b^2+c^2\right )}+\frac {e^{-i \sqrt {a^2+b^2+c^2} t} \left (e^{2 i \sqrt {a^2+b^2+c^2} t} a^2+a^2+2 b^2 e^{i \sqrt {a^2+b^2+c^2} t}+c^2 e^{2 i \sqrt {a^2+b^2+c^2} t}+c^2\right ) c_2}{2 \left (a^2+b^2+c^2\right )}-\frac {c e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (i \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t} a+i \sqrt {a^2+b^2+c^2} a+b c e^{i \sqrt {a^2+b^2+c^2} t}-b c\right ) c_3}{2 b \left (a^2+b^2+c^2\right )},z(t)\to -\frac {a e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (i \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t} b+i \sqrt {a^2+b^2+c^2} b+a c e^{i \sqrt {a^2+b^2+c^2} t}-a c\right ) c_1}{2 c \left (a^2+b^2+c^2\right )}-\frac {b e^{-i \sqrt {a^2+b^2+c^2} t} \left (-1+e^{i \sqrt {a^2+b^2+c^2} t}\right ) \left (-i \sqrt {a^2+b^2+c^2} e^{i \sqrt {a^2+b^2+c^2} t} a-i \sqrt {a^2+b^2+c^2} a+b c e^{i \sqrt {a^2+b^2+c^2} t}-b c\right ) c_2}{2 c \left (a^2+b^2+c^2\right )}+\frac {e^{-i \sqrt {a^2+b^2+c^2} t} \left (e^{2 i \sqrt {a^2+b^2+c^2} t} a^2+a^2+2 c^2 e^{i \sqrt {a^2+b^2+c^2} t}+b^2 e^{2 i \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 = 0.109 (sec), leaf count = 312 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}+{\it \_C2}\,\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) +{\it \_C3}\,\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) ,y \left ( t \right ) = -{\frac {1}{b \left ( {b}^{2}+{c}^{2} \right ) } \left ( \sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2 }}{\it \_C3}\,ac+\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C2}\,{a}^{2}b-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{\it \_C2}\,ac+\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C3}\,{a}^{2}b-{\it \_C1}\,{b}^{3}-{\it \_C1}\,b{c}^{2} \right ) },z \left ( t \right ) ={ \frac {1}{ \left ( {b}^{2}+{c}^{2} \right ) c} \left ( \sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{ \it \_C3}\,ab-\sin \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) { \it \_C2}\,{a}^{2}c-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}{\it \_C2}\,ab-\cos \left ( \sqrt {{a}^{2}+{b}^{2}+{c}^{2}}t \right ) {\it \_C3}\,{a}^{2}c+{\it \_C1}\,{b}^{2}c+{\it \_C1}\,{c}^{3} \right ) } \right \} \right \} \]