\[ \left \{x'(t)=x(t) \left (y(t)^2-z(t)^2\right ),y'(t)=y(t) \left (z(t)^2-x(t)^2\right ),z'(t)=z(t) \left (x(t)^2-y(t)^2\right )\right \} \] ✗ Mathematica : cpu = 0.0359181 (sec), leaf count = 0 , could not solve
DSolve[{Derivative[1][x][t] == x[t]*(y[t]^2 - z[t]^2), Derivative[1][y][t] == y[t]*(-x[t]^2 + z[t]^2), Derivative[1][z][t] == (x[t]^2 - y[t]^2)*z[t]}, {x[t], y[t], z[t]}, t]
✓ Maple : cpu = 8.43 (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 ) ={\frac { \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \left ( x \left ( t \right ) \right ) ^{3}}{ \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) x \left ( t \right ) -3\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}}} \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 ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \left ( x \left ( t \right ) \right ) ^{2}}\sqrt {- \left ( x \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) {\frac {{\rm d}^{3}}{{\rm d}{t}^{3}}}x \left ( t \right ) +4\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2} \left ( 1/4\,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) + \left ( x \left ( t \right ) \right ) ^{2} \left ( \left ( x \left ( t \right ) \right ) ^{3}+2\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \right ) x \left ( t \right ) }},y \left ( t \right ) ={\frac {1}{4\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \left ( x \left ( t \right ) \right ) ^{2}}\sqrt {- \left ( x \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) {\frac {{\rm d}^{3}}{{\rm d}{t}^{3}}}x \left ( t \right ) +4\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2} \left ( 1/4\,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) + \left ( x \left ( t \right ) \right ) ^{2} \left ( \left ( x \left ( t \right ) \right ) ^{3}+2\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \right ) x \left ( t \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 ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{4}{\it \_C1}\, \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{4}+ \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}{\it \_C1}\, \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \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 ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{4}{\it \_C1}\, \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \right ) ^{4}+ \left ( {{\rm e}^{{\it \_C1}\,t}} \right ) ^{2}{\it \_C1}\, \left ( {{\rm e}^{{\it \_C2}\,{\it \_C1}}} \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 \} \]