\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -y \left ( t \right ) ={{\rm e}^{t}},2\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +2\,y \left ( t \right ) =\cos \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.127016 (sec), leaf count = 122 \[ \left \{\left \{x(t)\to -\frac {3}{4} c_2 \left (e^{4 t}-1\right )+c_1+\frac {1}{68} e^{-4 t} \left (e^{4 t}-1\right ) \left (34 e^t+3 \sin (t)-12 \cos (t)\right )+\frac {1}{4} \left (2 e^{-3 t}+2 e^t+\frac {3}{17} e^{-4 t} \sin (t)+\sin (t)-\frac {12}{17} e^{-4 t} \cos (t)\right ),y(t)\to c_2 e^{4 t}+\frac {1}{51} \left (-34 e^t-3 \sin (t)+12 \cos (t)\right )\right \}\right \} \]
Maple: cpu = 0.078 (sec), leaf count = 47 \[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{t}}+{\frac {5\,\sin \left ( t \right ) }{17}}-{\frac {3\,\cos \left ( t \right ) }{17}}+{ \frac {{{\rm e}^{4\,t}}{\it \_C1}}{4}}+{\it \_C2},y \left ( t \right ) = -{\frac {2\,{{\rm e}^{t}}}{3}}+{\frac {4\,\cos \left ( t \right ) }{17}} -{\frac {\sin \left ( t \right ) }{17}}-{\frac {{{\rm e}^{4\,t}}{\it \_C1}}{3}} \right \} \right \} \]