\[ \left \{x'(t)=x(t) \left (-\left (x(t)^2+y(t)^2\right )\right )+x(t)+y(t),y'(t)=-y(t) \left (x(t)^2+y(t)^2\right )-x(t)+y(t)\right \} \] ✗ Mathematica : cpu = 0.108617 (sec), leaf count = 0 , could not solve
DSolve[{Derivative[1][x][t] == x[t] + y[t] - x[t]*(x[t]^2 + y[t]^2), Derivative[1][y][t] == -x[t] + y[t] - y[t]*(x[t]^2 + y[t]^2)}, {x[t], y[t]}, t]
✓ Maple : cpu = 4.945 (sec), leaf count = 249
\[ \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 \{ \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) {\it \_b} \left ( {\it \_a} \right ) -{\frac {1}{2\,{{\it \_a}}^{3}} \left ( 6\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{{\it \_a}}^{2}-4\,{\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{3}-4\,{{\it \_a}}^{4}-6\,{\it \_b} \left ( {\it \_a} \right ) {\it \_a}+4\,{{\it \_a}}^{2}-\sqrt {-64\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{{\it \_a}}^{6}+64\,{\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{7}-16\,{{\it \_a}}^{8}-64\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3}{{\it \_a}}^{3}+128\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{{\it \_a}}^{4}-48\,{\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{5}+48\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{{\it \_a}}^{2}-64\,{\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{3}+16\,{{\it \_a}}^{4}-12\,{\it \_b} \left ( {\it \_a} \right ) {\it \_a}+8\,{{\it \_a}}^{2}+1}+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 {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}- \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}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 \} \]