\[ \left \{x'(t)=c y(t)-b z(t),y'(t)=a z(t)-c x(t),z'(t)=b x(t)-a y(t)\right \} \] ✓ Mathematica : cpu = 0.044396 (sec), leaf count = 1445
DSolve[{Derivative[1][x][t] == c*y[t] - b*z[t], Derivative[1][y][t] == -(c*x[t]) + a*z[t], Derivative[1][z][t] == b*x[t] - a*y[t]},{x[t], y[t], z[t]},t]
\[\left \{\left \{x(t)\to \frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (2 e^{\sqrt {-a^2-b^2-c^2} t} a^2+b^2 e^{2 \sqrt {-a^2-b^2-c^2} t}+c^2 e^{2 \sqrt {-a^2-b^2-c^2} t}+b^2+c^2\right ) c_1}{2 \left (a^2+b^2+c^2\right )}-\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (a e^{\sqrt {-a^2-b^2-c^2} t} b-a b+c \sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t}+c \sqrt {-a^2-b^2-c^2}\right ) c_2}{2 \left (a^2+b^2+c^2\right )}-\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (-\sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t} b-\sqrt {-a^2-b^2-c^2} b+a c e^{\sqrt {-a^2-b^2-c^2} t}-a c\right ) c_3}{2 \left (a^2+b^2+c^2\right )},y(t)\to -\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (a e^{\sqrt {-a^2-b^2-c^2} t} b-a b-c \sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t}-c \sqrt {-a^2-b^2-c^2}\right ) c_1}{2 \left (a^2+b^2+c^2\right )}+\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (e^{2 \sqrt {-a^2-b^2-c^2} t} a^2+a^2+2 b^2 e^{\sqrt {-a^2-b^2-c^2} t}+c^2 e^{2 \sqrt {-a^2-b^2-c^2} t}+c^2\right ) c_2}{2 \left (a^2+b^2+c^2\right )}-\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (\sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t} a+\sqrt {-a^2-b^2-c^2} a+b c e^{\sqrt {-a^2-b^2-c^2} t}-b c\right ) c_3}{2 \left (a^2+b^2+c^2\right )},z(t)\to -\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (\sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t} b+\sqrt {-a^2-b^2-c^2} b+a c e^{\sqrt {-a^2-b^2-c^2} t}-a c\right ) c_1}{2 \left (a^2+b^2+c^2\right )}-\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (-1+e^{\sqrt {-a^2-b^2-c^2} t}\right ) \left (-\sqrt {-a^2-b^2-c^2} e^{\sqrt {-a^2-b^2-c^2} t} a-\sqrt {-a^2-b^2-c^2} a+b c e^{\sqrt {-a^2-b^2-c^2} t}-b c\right ) c_2}{2 \left (a^2+b^2+c^2\right )}+\frac {e^{-\sqrt {-a^2-b^2-c^2} t} \left (e^{2 \sqrt {-a^2-b^2-c^2} t} a^2+a^2+2 c^2 e^{\sqrt {-a^2-b^2-c^2} t}+b^2 e^{2 \sqrt {-a^2-b^2-c^2} t}+b^2\right ) c_3}{2 \left (a^2+b^2+c^2\right )}\right \}\right \}\] ✓ Maple : cpu = 0.1 (sec), leaf count = 257
dsolve({diff(x(t),t) = c*y(t)-b*z(t), diff(y(t),t) = a*z(t)-c*x(t), diff(z(t),t) = b*x(t)-a*y(t)})
\[\left \{x \left (t \right ) = c_{1}+c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right ), y \left (t \right ) = \frac {\left (-c_{3} a^{2} b +\sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a c \right ) \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+\left (-c_{2} a^{2} b -\sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a c \right ) \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+b c_{1} \left (b^{2}+c^{2}\right )}{a \left (b^{2}+c^{2}\right )}, z \left (t \right ) = \frac {\left (-c_{3} a^{2} c -\sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a b \right ) \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+\left (-c_{2} a^{2} c +\sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a b \right ) \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, t \right )+c c_{1} \left (b^{2}+c^{2}\right )}{a \left (b^{2}+c^{2}\right )}\right \}\]