\[ \left \{x'(t)=x(t) \left (x(t)^2+y(t)^2-1\right )-y(t),y'(t)=y(t) \left (x(t)^2+y(t)^2-1\right )+x(t)\right \} \] ✗ Mathematica : cpu = 0.0856214 (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 = 3.432 (sec), leaf count = 205
\[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) =0 \right \} ],[ \left \{ x \left ( t \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ {\frac {1}{2\,{{\it \_a}}^{3}} \left ( \sqrt {- \left ( 4\,{{\it \_a}}^{4}-4\,{{\it \_a}}^{2}-4\,{\it \_a}\,{\it \_b} \left ( {\it \_a} \right ) -1 \right ) \left ( 4\,{\it \_a}\,{\it \_b} \left ( {\it \_a} \right ) +2\,{{\it \_a}}^{2}+1 \right ) ^{2}}+2\, \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) {\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{3}-6\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{{\it \_a}}^{2}+ \left ( -4\,{{\it \_a}}^{3}-6\,{\it \_a} \right ) {\it \_b} \left ( {\it \_a} \right ) +4\,{{\it \_a}}^{4}-4\,{{\it \_a}}^{2}-1 \right ) }=0 \right \} , \left \{ {\it \_a}=x \left ( t \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right \} , \left \{ t=\int \! \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{\it \_C1},x \left ( t \right ) ={\it \_a} \right \} ] \right ) \right \} , \left \{ y \left ( t \right ) ={\frac { \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) \left ( x \left ( t \right ) \right ) ^{2}-3\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}x \left ( t \right ) -2\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) \left ( x \left ( t \right ) \right ) ^{2}+2\, \left ( x \left ( t \right ) \right ) ^{3}-{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -x \left ( t \right ) }{4\,x \left ( t \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +2\, \left ( x \left ( t \right ) \right ) ^{2}+1}} \right \} ] \right \} \]