\[ \left \{x'(t)=x(t) \left (y(t)^2-z(t)^2\right ),y'(t)=-y(t) \left (x(t)^2+z(t)^2\right ),z'(t)=z(t) \left (x(t)^2+y(t)^2\right )\right \} \] ✗ Mathematica : cpu = 0.0439705 (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 = 1.076 (sec), leaf count = 704
\[\left \{[\{x \left (t \right ) = 0\}, \{y \left (t \right ) = 0\}, \{z \left (t \right ) = c_{1}\}], \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) = \frac {\sqrt {c_{1} {\mathrm e}^{2 c_{1} c_{2}} {\mathrm e}^{2 c_{1} t}-c_{1} {\mathrm e}^{4 c_{1} c_{2}} {\mathrm e}^{4 c_{1} t}}}{{\mathrm e}^{2 c_{1} c_{2}} {\mathrm e}^{2 c_{1} t}-1}, y \left (t \right ) = -\frac {\sqrt {c_{1} {\mathrm e}^{2 c_{1} c_{2}} {\mathrm e}^{2 c_{1} t}-c_{1} {\mathrm e}^{4 c_{1} c_{2}} {\mathrm e}^{4 c_{1} t}}}{{\mathrm e}^{2 c_{1} c_{2}} {\mathrm e}^{2 c_{1} t}-1}\right \}, \left \{z \left (t \right ) = \frac {\sqrt {-\left (\frac {d}{d t}y \left (t \right )\right ) y \left (t \right )}}{y \left (t \right )}, z \left (t \right ) = -\frac {\sqrt {-\left (\frac {d}{d t}y \left (t \right )\right ) y \left (t \right )}}{y \left (t \right )}\right \}\right ], [\{x \left (t \right ) = c_{1}\}, \{y \left (t \right ) = i x \left (t \right ), y \left (t \right ) = -i x \left (t \right )\}, \{z \left (t \right ) = i x \left (t \right ), z \left (t \right ) = -i x \left (t \right )\}], \left [\left \{x \left (t \right ) = \frac {\sqrt {c_{1} {\mathrm e}^{2 c_{1} c_{2}} {\mathrm e}^{2 c_{1} t}-c_{1} {\mathrm e}^{4 c_{1} c_{2}} {\mathrm e}^{4 c_{1} t}}}{{\mathrm e}^{2 c_{1} c_{2}} {\mathrm e}^{2 c_{1} t}-1}, x \left (t \right ) = -\frac {\sqrt {c_{1} {\mathrm e}^{2 c_{1} c_{2}} {\mathrm e}^{2 c_{1} t}-c_{1} {\mathrm e}^{4 c_{1} c_{2}} {\mathrm e}^{4 c_{1} t}}}{{\mathrm e}^{2 c_{1} c_{2}} {\mathrm e}^{2 c_{1} t}-1}\right \}, \{y \left (t \right ) = 0\}, \left \{z \left (t \right ) = \frac {\sqrt {-\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )}}{x \left (t \right )}, z \left (t \right ) = -\frac {\sqrt {-\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )}}{x \left (t \right )}\right \}\right ], \left [\left \{x \left (t \right ) = \RootOf \left (c_{3}+t -\left (\int _{}^{\textit {\_Z}}-\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 c_{2} \textit {\_a}^{2}+16 c_{2}^{2}+c_{1}}\, \textit {\_a}}d \textit {\_a} \right )\right ), x \left (t \right ) = \RootOf \left (c_{3}+t -\left (\int _{}^{\textit {\_Z}}\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 c_{2} \textit {\_a}^{2}+16 c_{2}^{2}+c_{1}}\, \textit {\_a}}d \textit {\_a} \right )\right )\right \}, \left \{y \left (t \right ) = -\frac {\sqrt {-2 \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )-\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right ) x \left (t \right )}}{2 x \left (t \right )}, y \left (t \right ) = \frac {\sqrt {-2 \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )-\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right ) x \left (t \right )}}{2 x \left (t \right )}, y \left (t \right ) = -\frac {\sqrt {-2 \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )+\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right ) x \left (t \right )}}{2 x \left (t \right )}, y \left (t \right ) = \frac {\sqrt {-2 \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )+\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right ) x \left (t \right )}}{2 x \left (t \right )}\right \}, \left \{z \left (t \right ) = \frac {\sqrt {\left (x \left (t \right ) y \left (t \right )^{2}-\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )}}{x \left (t \right )}, z \left (t \right ) = -\frac {\sqrt {\left (x \left (t \right ) y \left (t \right )^{2}-\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )}}{x \left (t \right )}\right \}\right ]\right \}\]