\[ \left \{x'(t)=x(t) \left (y(t)^2-z(t)^2\right ),y'(t)=-y(t) \left (x(t)^2+z(t)^2\right ),z'(t)=z(t) \left (x(t)^2+y(t)^2\right )\right \} \] ✗ Mathematica : cpu = 0.0395462 (sec), leaf count = 0
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]
, 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 = 0.879 (sec), leaf count = 710
dsolve({diff(x(t),t) = x(t)*(y(t)^2-z(t)^2), diff(y(t),t) = -y(t)*(z(t)^2+x(t)^2), diff(z(t),t) = z(t)*(x(t)^2+y(t)^2)})
\[[\{x \left (t \right ) = 0\}, \{y \left (t \right ) = 0\}, \{z \left (t \right ) = c_{1}\}]\]