\[ \left \{\text {x1}'(t)=a \text {x2}(t)+b \text {x3}(t) \cos (c t)+b \text {x4}(t) \sin (c t),\text {x2}'(t)=-a \text {x1}(t)+b \text {x3}(t) \sin (c t)-b \text {x4}(t) \cos (c t),\text {x3}'(t)=a \text {x4}(t)-b \text {x1}(t) \cos (c t)-b \text {x2}(t) \sin (c t),\text {x4}'(t)=-a \text {x3}(t)-b \text {x1}(t) \sin (c t)+b \text {x2}(t) \cos (c t)\right \} \] ✓ Mathematica : cpu = 0.0109934 (sec), leaf count = 798
\[\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 = 1.156 (sec), leaf count = 2788
\[\left \{\left \{\mathit {x1} \left (t \right ) = c_{3} \sin \left (c t \right )+c_{4} \cos \left (c t \right )+c_{2}, \mathit {x2} \left (t \right ) = -c_{3} \cos \left (c t \right )+c_{4} \sin \left (c t \right )+c_{1}, \mathit {x3} \left (t \right ) = \frac {\left (c_{1} a \cos \left (c t \right )-c_{2} a \sin \left (c t \right )-c_{3} \left (a +c \right )\right ) b}{\left (a +c \right ) a}, \mathit {x4} \left (t \right ) = \frac {\left (c_{1} a \sin \left (c t \right )+c_{2} a \cos \left (c t \right )+c_{4} \left (a +c \right )\right ) b}{\left (a +c \right ) a}\right \}, \left \{\mathit {x1} \left (t \right ) = c_{1} {\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}+c_{3} {\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}, \mathit {x2} \left (t \right ) = \frac {-4 c_{1} \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}}}{4}+\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \left (a^{2}+a c +b^{2}+c^{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}+4 c_{2} \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}}}{4}+\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \left (a^{2}+a c +b^{2}+c^{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}-4 \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}}}{4}+\left (a^{2}+a c +b^{2}+c^{2}\right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\right ) \left (c_{3} {\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}-c_{4} {\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}\right )}{8 \left (a^{2}+a c +b^{2}\right ) c}, \mathit {x3} \left (t \right ) = \frac {8 c_{1} \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}} a \cos \left (c t \right )}{8}+\frac {\left (a^{3}+a \,b^{2}-b^{2} c \right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \cos \left (c t \right )}{2}+\left (\left (a +\frac {c}{2}\right ) c +\frac {\sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}{2}\right ) \left (a^{2}+a c +b^{2}\right ) \sin \left (c t \right )\right ) {\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}+8 c_{2} \left (-\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}} a \cos \left (c t \right )}{8}-\frac {\left (a^{3}+a \,b^{2}-b^{2} c \right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \cos \left (c t \right )}{2}+\left (\left (a +\frac {c}{2}\right ) c +\frac {\sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}{2}\right ) \left (a^{2}+a c +b^{2}\right ) \sin \left (c t \right )\right ) {\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}+8 c_{3} \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}} a \cos \left (c t \right )}{8}+\frac {\left (a^{3}+a \,b^{2}-b^{2} c \right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \cos \left (c t \right )}{2}+\left (\left (a +\frac {c}{2}\right ) c -\frac {\sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}{2}\right ) \left (a^{2}+a c +b^{2}\right ) \sin \left (c t \right )\right ) {\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}+8 c_{4} \left (-\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}} a \cos \left (c t \right )}{8}-\frac {\left (a^{3}+a \,b^{2}-b^{2} c \right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \cos \left (c t \right )}{2}+\left (\left (a +\frac {c}{2}\right ) c -\frac {\sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}{2}\right ) \left (a^{2}+a c +b^{2}\right ) \sin \left (c t \right )\right ) {\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}}{8 \left (a^{2}+a c +b^{2}\right ) b c}, \mathit {x4} \left (t \right ) = \frac {4 c_{1} \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}} a \sin \left (c t \right )}{4}-2 \left (\left (a +\frac {c}{2}\right ) c +\frac {\sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}{2}\right ) \left (a^{2}+a c +b^{2}\right ) \cos \left (c t \right )+\left (a^{3}+a \,b^{2}-b^{2} c \right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \sin \left (c t \right )\right ) {\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}-4 c_{2} \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}} a \sin \left (c t \right )}{4}+2 \left (\left (a +\frac {c}{2}\right ) c +\frac {\sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}{2}\right ) \left (a^{2}+a c +b^{2}\right ) \cos \left (c t \right )+\left (a^{3}+a \,b^{2}-b^{2} c \right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \sin \left (c t \right )\right ) {\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}+4 c_{3} \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}} a \sin \left (c t \right )}{4}-2 \left (\left (a +\frac {c}{2}\right ) c -\frac {\sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}{2}\right ) \left (a^{2}+a c +b^{2}\right ) \cos \left (c t \right )+\left (a^{3}+a \,b^{2}-b^{2} c \right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \sin \left (c t \right )\right ) {\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}-4 c_{4} \left (\frac {\left (-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}\right )^{\frac {3}{2}} a \sin \left (c t \right )}{4}+2 \left (\left (a +\frac {c}{2}\right ) c -\frac {\sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}{2}\right ) \left (a^{2}+a c +b^{2}\right ) \cos \left (c t \right )+\left (a^{3}+a \,b^{2}-b^{2} c \right ) \sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, \sin \left (c t \right )\right ) {\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {\left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right ) c^{2}}}\, t}{2}}}{8 \left (a^{2}+a c +b^{2}\right ) b c}\right \}\right \}\]