\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =x \left ( t \right ) +y \left ( t \right ) -x \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2} \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =-x \left ( t \right ) +y \left ( t \right ) -y \left ( t \right ) \left ( \left ( x \left ( t \right ) \right ) ^{2}+ \left ( y \left ( t \right ) \right ) ^{2} \right ) \right \} } \]
Mathematica: cpu = 0.107014 (sec), leaf count = 56 \[ \text {DSolve}\left [\left \{x'(t)=x(t) \left (-\left (x(t)^2+y(t)^2\right )\right )+x(t)+y(t),y'(t)=-y(t) \left (x(t)^2+y(t)^2\right )-x(t)+y(t)\right \},\{x(t),y(t)\},t\right ] \]
Maple: cpu = 2.278 (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 \_a}}^ {3}{\it \_b} \left ( {\it \_a} \right ) -4\,{{\it \_a}}^{4}-6\,{\it \_a} \,{\it \_b} \left ( {\it \_a} \right ) +4\,{{\it \_a}}^{2}+1-\sqrt {-64 \,{{\it \_a}}^{6} \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^ {2}+64\,{\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{7}-16\,{{\it \_a}}^{8}-64\,{{\it \_a}}^{3} \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3}+128\,{{\it \_a}}^{4} \left ( {\it \_b} \left ( { \it \_a} \right ) \right ) ^{2}-48\,{{\it \_a}}^{5}{\it \_b} \left ( { \it \_a} \right ) +48\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{{\it \_a}}^{2}-64\,{{\it \_a}}^{3}{\it \_b} \left ( {\it \_a} \right ) +16\,{{\it \_a}}^{4}-12\,{\it \_a}\,{\it \_b} \left ( { \it \_a} \right ) +8\,{{\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}-2\, \left ( x \left ( t \right ) \right ) ^{3}-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 \} \]