\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =x \left ( t \right ) f \left ( t \right ) +y \left ( t \right ) g \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =-x \left ( t \right ) g \left ( t \right ) +y \left ( t \right ) f \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.062508 (sec), leaf count = 107 \[ \left \{\left \{x(t)\to c_2 e^{\int _1^t f(K[2]) \, dK[2]} \sin \left (\int _1^t g(K[1]) \, dK[1]\right )+c_1 e^{\int _1^t f(K[2]) \, dK[2]} \cos \left (\int _1^t g(K[1]) \, dK[1]\right ),y(t)\to c_2 e^{\int _1^t f(K[2]) \, dK[2]} \cos \left (\int _1^t g(K[1]) \, dK[1]\right )-c_1 e^{\int _1^t f(K[2]) \, dK[2]} \sin \left (\int _1^t g(K[1]) \, dK[1]\right )\right \}\right \} \]
Maple: cpu = 0.296 (sec), leaf count = 57 \[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{\int \!\tan \left ( { \it \_C1}-\int \!g \left ( t \right ) \,{\rm d}t \right ) g \left ( t \right ) +f \left ( t \right ) \,{\rm d}t}}{\it \_C2},y \left ( t \right ) =\tan \left ( {\it \_C1}-\int \!g \left ( t \right ) \,{\rm d}t \right ) {{\rm e}^{\int \!\tan \left ( {\it \_C1}-\int \!g \left ( t \right ) \,{\rm d}t \right ) g \left ( t \right ) +f \left ( t \right ) \,{\rm d}t}}{\it \_C2} \right \} \right \} \]