\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =-x \left ( t \right ) \left ( x \left ( t \right ) +y \left ( t \right ) \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =y \left ( t \right ) \left ( x \left ( t \right ) +y \left ( t \right ) \right ) \right \} } \]
Mathematica: cpu = 0.032004 (sec), leaf count = 64 \[ \left \{\left \{y(t)\to -\sqrt {c_1} \cot \left (\sqrt {c_1} t-\sqrt {c_1} c_2\right ),x(t)\to -\sqrt {c_1} \tan \left (\sqrt {c_1} t-\sqrt {c_1} c_2\right )\right \}\right \} \]
Maple: cpu = 0.109 (sec), leaf count = 54 \[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) = \left ( {\it \_C1}-t \right ) ^{-1} \right \} ],[ \left \{ x \left ( t \right ) ={\frac {1}{{\it \_C1}}\tanh \left ( {\frac {{\it \_C2}+t}{{\it \_C1}}} \right ) } \right \} , \left \{ y \left ( t \right ) =-{\frac { \left ( x \left ( t \right ) \right ) ^{2}+{\frac {\rm d}{ {\rm d}t}}x \left ( t \right ) }{x \left ( t \right ) }} \right \} ] \right \} \]