\[ \left \{x'(t)=x(t) \left (x(t)^2+y(t)^2-1\right )-y(t),y'(t)=y(t) \left (x(t)^2+y(t)^2-1\right )+x(t)\right \} \] ✗ Mathematica : cpu = 0.236862 (sec), leaf count = 0 , could not solve
DSolve[{Derivative[1][x][t] == -y[t] + x[t]*(-1 + x[t]^2 + y[t]^2), Derivative[1][y][t] == x[t] + y[t]*(-1 + x[t]^2 + y[t]^2)}, {x[t], y[t]}, t]
✓ Maple : cpu = 3.939 (sec), leaf count = 202
\[\left \{[\{x \left (t \right ) = 0\}, \{y \left (t \right ) = 0\}], \left [\left \{x \left (t \right ) = \mathit {ODESolStruc} \left (\textit {\_a} , \left [\left \{\textit {\_}b\left (\textit {\_a} \right ) \left (\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )\right )+\frac {4 \textit {\_a}^{4}-4 \textit {\_a}^{3} \textit {\_}b\left (\textit {\_a} \right )-6 \textit {\_a}^{2} \textit {\_}b\left (\textit {\_a} \right )^{2}-4 \textit {\_a}^{2}-6 \textit {\_a} \textit {\_}b\left (\textit {\_a} \right )+\sqrt {-\left (4 \textit {\_a}^{4}-4 \textit {\_a}^{2}-4 \textit {\_a} \textit {\_}b\left (\textit {\_a} \right )-1\right ) \left (2 \textit {\_a}^{2}+4 \textit {\_a} \textit {\_}b\left (\textit {\_a} \right )+1\right )^{2}}-1}{2 \textit {\_a}^{3}}=0\right \}, \left \{\textit {\_a} =x \left (t \right ), \textit {\_}b\left (\textit {\_a} \right )=\frac {d}{d t}x \left (t \right )\right \}, \left \{t =c_{1}+\int \frac {1}{\textit {\_}b\left (\textit {\_a} \right )}d \textit {\_a} , x \left (t \right )=\textit {\_a} \right \}\right ]\right )\right \}, \left \{y \left (t \right ) = \frac {\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )^{2}-3 \left (\frac {d}{d t}x \left (t \right )\right )^{2} x \left (t \right )-2 \left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )^{2}+2 x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )-x \left (t \right )}{4 \left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )+2 x \left (t \right )^{2}+1}\right \}\right ]\right \}\]