\[ \left \{x'(t)=f(t) x(t)+g(t) y(t),y'(t)=f(t) y(t)-g(t) x(t)\right \} \] ✓ Mathematica : cpu = 0.140816 (sec), leaf count = 105
\[\left \{\left \{x(t)\to e^{\text {Integrate}[f(K[2]),\{K[2],1,t\},\text {Assumptions}\to \text {True}]} \left (c_2 \sin (\text {Integrate}[g(K[1]),\{K[1],1,t\},\text {Assumptions}\to \text {True}])+c_1 \cos (\text {Integrate}[g(K[1]),\{K[1],1,t\},\text {Assumptions}\to \text {True}])\right ),y(t)\to e^{\text {Integrate}[f(K[2]),\{K[2],1,t\},\text {Assumptions}\to \text {True}]} \left (c_2 \cos (\text {Integrate}[g(K[1]),\{K[1],1,t\},\text {Assumptions}\to \text {True}])-c_1 \sin (\text {Integrate}[g(K[1]),\{K[1],1,t\},\text {Assumptions}\to \text {True}])\right )\right \}\right \}\]
✓ Maple : cpu = 0.481 (sec), leaf count = 57
\[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{\int \!\tan \left ( {\it \_C1}-\int \!g \left ( t \right ) \,{\rm d}t \right ) g \left ( t \right ) +f \left ( t \right ) \,{\rm d}t}}{\it \_C2},y \left ( t \right ) =\tan \left ( {\it \_C1}-\int \!g \left ( t \right ) \,{\rm d}t \right ) {{\rm e}^{\int \!\tan \left ( {\it \_C1}-\int \!g \left ( t \right ) \,{\rm d}t \right ) g \left ( t \right ) +f \left ( t \right ) \,{\rm d}t}}{\it \_C2} \right \} \right \} \]