\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =x \left ( t \right ) \left ( \left ( y \left ( t \right ) \right ) ^{2}- \left ( z \left ( t \right ) \right ) ^{2} \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =y \left ( t \right ) \left ( \left ( z \left ( t \right ) \right ) ^{2}- \left ( x \left ( t \right ) \right ) ^{2} \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =z \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}- \left ( y \left ( t \right ) \right ) ^{2} \right ) \right \} } \]
Mathematica: cpu = 0.050506 (sec), leaf count = 67 \[ \text {DSolve}\left [\left \{x'(t)=x(t) \left (y(t)^2-z(t)^2\right ),y'(t)=y(t) \left (z(t)^2-x(t)^2\right ),z'(t)=z(t) \left (x(t)^2-y(t)^2\right )\right \},\{x(t),y(t),z(t)\},t\right ] \]
Maple: cpu = 1.357 (sec), leaf count = 741 \[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) =0 \right \} , \left \{ z \left ( t \right ) ={\it \_C1} \right \} ],[ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) ={\frac {1}{ \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1} }} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}-1}\sqrt { \left ( \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}-1 \right ) {\it \_C1}\, \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( { {\rm e}^{{\it \_C1}\,t}} \right ) ^{2}}},y \left ( t \right ) =-{\frac {1 }{ \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( { {\rm e}^{{\it \_C1}\,t}} \right ) ^{2}-1}\sqrt { \left ( \left ( { {\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}-1 \right ) {\it \_C1}\, \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}}} \right \} , \left \{ z \left ( t \right ) ={\frac {1}{y \left ( t \right ) }\sqrt {y \left ( t \right ) {\frac {\rm d}{{\rm d}t}} y \left ( t \right ) }},z \left ( t \right ) =-{\frac {1}{y \left ( t \right ) }\sqrt {y \left ( t \right ) {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) }} \right \} ],[ \left \{ x \left ( t \right ) ={\it \_C1} \right \} , \left \{ y \left ( t \right ) =-x \left ( t \right ) \right \} , \left \{ z \left ( t \right ) =-x \left ( t \right ) \right \} ],[ \left \{ x \left ( t \right ) ={\it \_C1} \right \} , \left \{ y \left ( t \right ) =-x \left ( t \right ) \right \} , \left \{ z \left ( t \right ) = x \left ( t \right ) \right \} ],[ \left \{ x \left ( t \right ) ={\it \_C1 } \right \} , \left \{ y \left ( t \right ) =x \left ( t \right ) \right \} , \left \{ z \left ( t \right ) =-x \left ( t \right ) \right \} ],[ \left \{ x \left ( t \right ) ={\it \_C1} \right \} , \left \{ y \left ( t \right ) =x \left ( t \right ) \right \} , \left \{ z \left ( t \right ) =x \left ( t \right ) \right \} ],[ \left \{ x \left ( t \right ) ={\it ODESolStruc} \left ( {{\rm e}^{\int \!{\it \_g} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it \_C2}}},[ \left \{ {\frac {\rm d}{ {\rm d}{\it \_f}}}{\it \_g} \left ( {\it \_f} \right ) =4\,{\frac { \left ( {\it \_g} \left ( {\it \_f} \right ) \right ) ^{3}}{{\it \_f}} \left ( 3\,{{\it \_f}}^{2}+2\,\sqrt {{\frac {3\,{{\it \_f}}^{2}{\it \_g} \left ( {\it \_f} \right ) +{\it \_g} \left ( {\it \_f} \right ) +{ \it \_f}}{{\it \_g} \left ( {\it \_f} \right ) }}}+1 \right ) }+10\, \left ( {\it \_g} \left ( {\it \_f} \right ) \right ) ^{2}+{\frac {{\it \_g} \left ( {\it \_f} \right ) }{{\it \_f}}} \right \} , \left \{ {\it \_f}={\frac {{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }{ \left ( x \left ( t \right ) \right ) ^{3}}},{\it \_g} \left ( {\it \_f} \right ) = -{ \left ( x \left ( t \right ) \right ) ^{3} \left ( -{\frac { \left ( { \frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) x \left ( t \right ) }{{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }}+3\, {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{-1}} \right \} , \left \{ t=\int \!{\frac {{\it \_g} \left ( {\it \_f} \right ) }{{\it \_f}\, \left ( {{\rm e}^{\int \!{\it \_g} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it \_C2}}} \right ) ^{2}}}\,{\rm d}{\it \_f}+{\it \_C1},x \left ( t \right ) ={{\rm e}^{\int \!{\it \_g} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it \_C2}}} \right \} ] \right ) \right \} , \left \{ y \left ( t \right ) =-{\frac {1}{4\, \left ( x \left ( t \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) } \sqrt {x \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \left ( 4\, \left ( x \left ( t \right ) \right ) ^{5}{ \frac {\rm d}{{\rm d}t}}x \left ( t \right ) +8\, \left ( x \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}- \left ( {\frac {{\rm d}^{3}}{{\rm d}{t}^{3}}}x \left ( t \right ) \right ) x \left ( t \right ) + \left ( {\frac {{\rm d} ^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) {\frac {\rm d}{ {\rm d}t}}x \left ( t \right ) \right ) }},y \left ( t \right ) ={\frac {1 }{4\, \left ( x \left ( t \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}t}} x \left ( t \right ) }\sqrt {x \left ( t \right ) \left ( {\frac {\rm d}{ {\rm d}t}}x \left ( t \right ) \right ) \left ( 4\, \left ( x \left ( t \right ) \right ) ^{5}{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +8\, \left ( x \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d} t}}x \left ( t \right ) \right ) ^{2}- \left ( {\frac {{\rm d}^{3}}{ {\rm d}{t}^{3}}}x \left ( t \right ) \right ) x \left ( t \right ) + \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) }} \right \} , \left \{ z \left ( t \right ) ={\frac {1}{x \left ( t \right ) }\sqrt {x \left ( t \right ) \left ( x \left ( t \right ) \left ( y \left ( t \right ) \right ) ^{2}-{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) }},z \left ( t \right ) =-{\frac {1}{x \left ( t \right ) }\sqrt {x \left ( t \right ) \left ( x \left ( t \right ) \left ( y \left ( t \right ) \right ) ^{2}-{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) }} \right \} ],[ \left \{ x \left ( t \right ) ={\frac {1}{ \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}-1}\sqrt {- \left ( \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( { {\rm e}^{{\it \_C1}\,t}} \right ) ^{2}-1 \right ) {\it \_C1}\, \left ( { {\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}}},x \left ( t \right ) =-{\frac {1}{ \left ( { {\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}-1}\sqrt {- \left ( \left ( {{\rm e}^{{\it \_C2} \,{\it \_C1}}} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^ {2}-1 \right ) {\it \_C1}\, \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{2} \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}}} \right \} , \left \{ y \left ( t \right ) =0 \right \} , \left \{ z \left ( t \right ) ={\frac {1}{x \left ( t \right ) }\sqrt {-x \left ( t \right ) { \frac {\rm d}{{\rm d}t}}x \left ( t \right ) }},z \left ( t \right ) =-{ \frac {1}{x \left ( t \right ) }\sqrt {-x \left ( t \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }} \right \} ] \right \} \]