\[ \left \{\text {x1}'(t)=a \text {x2}(t)+b \text {x3}(t) \cos (c t)+b \text {x4}(t) \sin (c t),\text {x2}'(t)=-a \text {x1}(t)+b \text {x3}(t) \sin (c t)-b \text {x4}(t) \cos (c t),\text {x3}'(t)=a \text {x4}(t)-b \text {x1}(t) \cos (c t)-b \text {x2}(t) \sin (c t),\text {x4}'(t)=-a \text {x3}(t)-b \text {x1}(t) \sin (c t)+b \text {x2}(t) \cos (c t)\right \} \] ✓ Mathematica : cpu = 0.0099564 (sec), leaf count = 798
DSolve[{Derivative[1][x1][t] == a*x2[t] + b*Cos[c*t]*x3[t] + b*Sin[c*t]*x4[t], Derivative[1][x2][t] == -(a*x1[t]) + b*Sin[c*t]*x3[t] - b*Cos[c*t]*x4[t], Derivative[1][x3][t] == -(b*Cos[c*t]*x1[t]) - b*Sin[c*t]*x2[t] + a*x4[t], Derivative[1][x4][t] == -(b*Sin[c*t]*x1[t]) + b*Cos[c*t]*x2[t] - a*x3[t]},{x1[t], x2[t], x3[t], x4[t]},t]
\[\left \{\left \{\text {x1}(t)\to c_3 \cos \left (\left (\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )+c_1 \cos \left (\left (\frac {c}{2}+\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )+c_4 \sin \left (\left (\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )+c_2 \sin \left (\left (\frac {c}{2}+\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right ),\text {x2}(t)\to -c_4 \cos \left (\left (\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )-c_2 \cos \left (\left (\frac {c}{2}+\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )+c_3 \sin \left (\left (\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )+c_1 \sin \left (\left (\frac {c}{2}+\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right ),\text {x3}(t)\to \frac {\left (a+\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) c_4 \cos \left (\frac {1}{2} \left (c+\sqrt {4 a^2+4 c a+4 b^2+c^2}\right ) t\right )}{b}+\frac {\left (a+\frac {c}{2}+\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) c_2 \cos \left (\left (\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )}{b}+\frac {\left (a+\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) c_3 \sin \left (\frac {1}{2} \left (c+\sqrt {4 a^2+4 c a+4 b^2+c^2}\right ) t\right )}{b}+\frac {\left (a+\frac {c}{2}+\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) c_1 \sin \left (\left (\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )}{b},\text {x4}(t)\to -\frac {\left (a+\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) c_3 \cos \left (\frac {1}{2} \left (c+\sqrt {4 a^2+4 c a+4 b^2+c^2}\right ) t\right )}{b}-\frac {\left (a+\frac {c}{2}+\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) c_1 \cos \left (\left (\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )}{b}+\frac {\left (a+\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) c_4 \sin \left (\frac {1}{2} \left (c+\sqrt {4 a^2+4 c a+4 b^2+c^2}\right ) t\right )}{b}+\frac {\left (a+\frac {c}{2}+\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) c_2 \sin \left (\left (\frac {c}{2}-\frac {1}{2} \sqrt {4 b^2+(2 a+c)^2}\right ) t\right )}{b}\right \}\right \}\] ✓ Maple : cpu = 1.161 (sec), leaf count = 2788
dsolve({diff(x1(t),t) = a*x2(t)+b*x3(t)*cos(c*t)+b*x4(t)*sin(c*t), diff(x2(t),t) = -a*x1(t)+b*x3(t)*sin(c*t)-b*x4(t)*cos(c*t), diff(x3(t),t) = -b*x1(t)*cos(c*t)-b*x2(t)*sin(c*t)+a*x4(t), diff(x4(t),t) = -b*x1(t)*sin(c*t)+b*x2(t)*cos(c*t)-a*x3(t)})
\[\left \{\mathit {x1} \left (t \right ) = c_{2}+c_{3} \sin \left (c t \right )+c_{4} \cos \left (c t \right ), \mathit {x2} \left (t \right ) = -\cos \left (c t \right ) c_{3}+\sin \left (c t \right ) c_{4}+c_{1}, \mathit {x3} \left (t \right ) = \frac {b \left (\cos \left (c t \right ) c_{1} a -\sin \left (c t \right ) c_{2} a -c_{3} \left (a +c \right )\right )}{a \left (a +c \right )}, \mathit {x4} \left (t \right ) = \frac {b \left (\cos \left (c t \right ) c_{2} a +\sin \left (c t \right ) c_{1} a +c_{4} \left (a +c \right )\right )}{a \left (a +c \right )}\right \}\]