\[ \left \{(x(t)-y(t)) (x(t)-z(t)) x'(t)=f(t),(y(t)-x(t)) (y(t)-z(t)) y'(t)=f(t),(z(t)-x(t)) (z(t)-y(t)) z'(t)=f(t)\right \} \] ✓ Mathematica : cpu = 0.263005 (sec), leaf count = 2168
\[\left \{\left \{x(t)\to \frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}},y(t)\to \frac {1}{2} \left (\frac {2 c_1}{3}-\sqrt {\left (-\frac {2 c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ){}^2-4 \left (\left (\frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ){}^2-c_1 \left (\frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right )+c_2\right )}-\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ),z(t)\to -\frac {\sqrt [3]{2} c_1{}^2}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}+\frac {c_1{}^2}{3\ 2^{2/3} \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}+\frac {c_1}{3}+\frac {1}{2} \sqrt {\left (-\frac {2 c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ){}^2-4 \left (\left (\frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ){}^2-c_1 \left (\frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right )+c_2\right )}-\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{6 \sqrt [3]{2}}+\frac {\sqrt [3]{2} c_2}{\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}-\frac {c_2}{2^{2/3} \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right \}\right \}\] ✓ Maple : cpu = 1.512 (sec), leaf count = 899
\[\left [\left \{x \relax (t ) = \int \frac {3 \left (\left (-i \sqrt {3}-1\right ) \left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {2}{3}}+\left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right ) c_{1} \left (i \sqrt {3}-1\right )\right ) f \relax (t )}{\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )}d t +c_{3}, x \relax (t ) = \int -\frac {3 f \relax (t ) \left (\left (1-i \sqrt {3}\right ) \left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {2}{3}}+\left (1+i \sqrt {3}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right ) c_{1}\right )}{\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )}d t +c_{3}, x \relax (t ) = \int \frac {6 f \relax (t ) \left (c_{1}^{4}+11664 \left (\int f \relax (t )d t \right )^{2} c_{1}-23328 \left (\int f \relax (t )d t \right ) c_{1} c_{2}+11664 c_{1} c_{2}^{2}+\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {2}{3}}\right )}{\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )}d t +c_{3}\right \}, \left \{y \relax (t ) = \frac {4 \left (\frac {d}{d t}x \relax (t )\right )^{3} x \relax (t )+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) f \relax (t )-\left (\frac {d}{d t}f \relax (t )\right ) \left (\frac {d}{d t}x \relax (t )\right )-\sqrt {-16 \left (\frac {d}{d t}x \relax (t )\right )^{5} f \relax (t )+\left (\frac {d}{d t}x \relax (t )\right )^{2} \left (\frac {d}{d t}f \relax (t )\right )^{2}-2 \left (\frac {d}{d t}x \relax (t )\right ) \left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) f \relax (t ) \left (\frac {d}{d t}f \relax (t )\right )+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right )^{2} f \relax (t )^{2}}}{4 \left (\frac {d}{d t}x \relax (t )\right )^{3}}, y \relax (t ) = \frac {4 \left (\frac {d}{d t}x \relax (t )\right )^{3} x \relax (t )+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) f \relax (t )-\left (\frac {d}{d t}f \relax (t )\right ) \left (\frac {d}{d t}x \relax (t )\right )+\sqrt {-16 \left (\frac {d}{d t}x \relax (t )\right )^{5} f \relax (t )+\left (\frac {d}{d t}x \relax (t )\right )^{2} \left (\frac {d}{d t}f \relax (t )\right )^{2}-2 \left (\frac {d}{d t}x \relax (t )\right ) \left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) f \relax (t ) \left (\frac {d}{d t}f \relax (t )\right )+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right )^{2} f \relax (t )^{2}}}{4 \left (\frac {d}{d t}x \relax (t )\right )^{3}}\right \}, \left \{z \relax (t ) = \frac {x \relax (t ) \left (x \relax (t )-y \relax (t )\right ) \left (\frac {d}{d t}x \relax (t )\right )-f \relax (t )}{\left (\frac {d}{d t}x \relax (t )\right ) \left (x \relax (t )-y \relax (t )\right )}\right \}\right ]\]