\[ \left \{x'(t)=x(t) \left (-\left (x(t)^2+y(t)^2\right )\right )+x(t)+y(t),y'(t)=-y(t) \left (x(t)^2+y(t)^2\right )-x(t)+y(t)\right \} \] ✗ Mathematica : cpu = 0.111331 (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 = 4.011 (sec), leaf count = 203
\[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) =0 \right \} ],[ \left \{ x \left ( t \right ) ={\it ODESolStruc} \left ( {\it \_a},[ \left \{ {\frac {1}{2\,{{\it \_a}}^{3}} \left ( \sqrt {- \left ( 4\,{{\it \_a}}^{4}+4\,{\it \_a}\,{\it \_b} \left ( {\it \_a} \right ) -4\,{{\it \_a}}^{2}-1 \right ) \left ( 2\,{{\it \_a}}^{2}-4\,{\it \_a}\,{\it \_b} \left ( {\it \_a} \right ) +1 \right ) ^{2}}+2\, \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) {\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{3}-6\,{{\it \_a}}^{2} \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}+ \left ( 4\,{{\it \_a}}^{3}+6\,{\it \_a} \right ) {\it \_b} \left ( {\it \_a} \right ) +4\,{{\it \_a}}^{4}-4\,{{\it \_a}}^{2}-1 \right ) }=0 \right \} , \left \{ {\it \_a}=x \left ( t \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right \} , \left \{ t=\int \! \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{\it \_C1},x \left ( t \right ) ={\it \_a} \right \} ] \right ) \right \} , \left \{ y \left ( t \right ) ={\frac {2\, \left ( x \left ( t \right ) \right ) ^{3}+ \left ( x \left ( t \right ) \right ) ^{2}{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +2\, \left ( x \left ( t \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -3\,x \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}-x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }{2\, \left ( x \left ( t \right ) \right ) ^{2}-4\,x \left ( t \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +1}} \right \} ] \right \} \]