\[ \left \{x'(t)=x(t) \left (y(t)^2-z(t)^2\right ),y'(t)=-y(t) \left (x(t)^2+z(t)^2\right ),z'(t)=z(t) \left (x(t)^2+y(t)^2\right )\right \} \] ✗ Mathematica : cpu = 0.0346893 (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 = 1.534 (sec), leaf count = 704
\[ \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 ( {{\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}}},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 ( {{\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 \{ 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 ) =-ix \left ( t \right ) ,y \left ( t \right ) =ix \left ( t \right ) \right \} , \left \{ z \left ( t \right ) =-ix \left ( t \right ) ,z \left ( t \right ) =ix \left ( t \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 \} ],[ \left \{ x \left ( t \right ) ={\it RootOf} \left ( -\int ^{{\it \_Z}}\!-2\,{\frac {1}{\sqrt {4\,{{\it \_a}}^{4}-16\,{\it \_C2}\,{{\it \_a}}^{2}+16\,{{\it \_C2}}^{2}+{\it \_C1}}{\it \_a}}}{d{\it \_a}}+t+{\it \_C3} \right ) ,x \left ( t \right ) ={\it RootOf} \left ( -\int ^{{\it \_Z}}\!2\,{\frac {1}{\sqrt {4\,{{\it \_a}}^{4}-16\,{\it \_C2}\,{{\it \_a}}^{2}+16\,{{\it \_C2}}^{2}+{\it \_C1}}{\it \_a}}}{d{\it \_a}}+t+{\it \_C3} \right ) \right \} , \left \{ y \left ( t \right ) =-{\frac {1}{2\,x \left ( t \right ) }\sqrt {-2\,x \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{3}-{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -\sqrt { \left ( x \left ( t \right ) \right ) ^{6}- \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) x \left ( t \right ) +2\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}} \right ) }},y \left ( t \right ) ={\frac {1}{2\,x \left ( t \right ) }\sqrt {-2\,x \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{3}-{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -\sqrt { \left ( x \left ( t \right ) \right ) ^{6}- \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) x \left ( t \right ) +2\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}} \right ) }},y \left ( t \right ) =-{\frac {1}{2\,x \left ( t \right ) }\sqrt {-2\,x \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{3}-{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +\sqrt { \left ( x \left ( t \right ) \right ) ^{6}- \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) x \left ( t \right ) +2\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}} \right ) }},y \left ( t \right ) ={\frac {1}{2\,x \left ( t \right ) }\sqrt {-2\,x \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{3}-{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +\sqrt { \left ( x \left ( t \right ) \right ) ^{6}- \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) x \left ( t \right ) +2\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}} \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 \} ] \right \} \]