\[ \left \{x'(t)=y(t)^2-\cos (x(t)),y'(t)=y(t) (-\sin (x(t)))\right \} \] ✓ Mathematica : cpu = 216.129 (sec), leaf count = 3406
\[\left \{\left \{y(t)\to \frac {3 \sqrt [3]{2} \cos \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}\right ){}^{2/3}}{2\ 2^{2/3} \cos ^2(K[1])+2 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}\right ){}^{2/3} \cos (K[1])+3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}}+\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[1])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}}}dK[1]\& \right ]\left [\frac {t}{2}+c_2\right ]\right )}{\sqrt [3]{81 c_1+\sqrt {6561 c_1{}^2-2916 \cos ^3\left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}\right ){}^{2/3}}{2\ 2^{2/3} \cos ^2(K[1])+2 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}\right ){}^{2/3} \cos (K[1])+3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}}+\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[1])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}}}dK[1]\& \right ]\left [\frac {t}{2}+c_2\right ]\right )}}}+\frac {\sqrt [3]{81 c_1+\sqrt {6561 c_1{}^2-2916 \cos ^3\left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}\right ){}^{2/3}}{2\ 2^{2/3} \cos ^2(K[1])+2 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}\right ){}^{2/3} \cos (K[1])+3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}}+\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[1])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}}}dK[1]\& \right ]\left [\frac {t}{2}+c_2\right ]\right )}}}{3 \sqrt [3]{2}},x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}\right ){}^{2/3}}{2\ 2^{2/3} \cos ^2(K[1])+2 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}\right ){}^{2/3} \cos (K[1])+3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}}+\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[1])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[1])}}}dK[1]\& \right ]\left [\frac {t}{2}+c_2\right ]\right \},\left \{y(t)\to -\frac {3 \left (1+i \sqrt {3}\right ) \cos \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}\right ){}^{2/3}}{2 i 2^{2/3} \sqrt {3} \cos ^2(K[2])-2\ 2^{2/3} \cos ^2(K[2])+4 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}\right ){}^{2/3} \cos (K[2])-3 i \sqrt [3]{2} \sqrt {3} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-i \sqrt [3]{2} \sqrt {3} \sqrt {9 c_1{}^2-4 \cos ^3(K[2])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[2])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}}dK[2]\& \right ]\left [\frac {t}{4}+c_2\right ]\right )}{2^{2/3} \sqrt [3]{81 c_1+\sqrt {6561 c_1{}^2-2916 \cos ^3\left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}\right ){}^{2/3}}{2 i 2^{2/3} \sqrt {3} \cos ^2(K[2])-2\ 2^{2/3} \cos ^2(K[2])+4 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}\right ){}^{2/3} \cos (K[2])-3 i \sqrt [3]{2} \sqrt {3} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-i \sqrt [3]{2} \sqrt {3} \sqrt {9 c_1{}^2-4 \cos ^3(K[2])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[2])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}}dK[2]\& \right ]\left [\frac {t}{4}+c_2\right ]\right )}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{81 c_1+\sqrt {6561 c_1{}^2-2916 \cos ^3\left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}\right ){}^{2/3}}{2 i 2^{2/3} \sqrt {3} \cos ^2(K[2])-2\ 2^{2/3} \cos ^2(K[2])+4 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}\right ){}^{2/3} \cos (K[2])-3 i \sqrt [3]{2} \sqrt {3} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-i \sqrt [3]{2} \sqrt {3} \sqrt {9 c_1{}^2-4 \cos ^3(K[2])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[2])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}}dK[2]\& \right ]\left [\frac {t}{4}+c_2\right ]\right )}}}{6 \sqrt [3]{2}},x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}\right ){}^{2/3}}{2 i 2^{2/3} \sqrt {3} \cos ^2(K[2])-2\ 2^{2/3} \cos ^2(K[2])+4 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}\right ){}^{2/3} \cos (K[2])-3 i \sqrt [3]{2} \sqrt {3} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-i \sqrt [3]{2} \sqrt {3} \sqrt {9 c_1{}^2-4 \cos ^3(K[2])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}-\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[2])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[2])}}}dK[2]\& \right ]\left [\frac {t}{4}+c_2\right ]\right \},\left \{y(t)\to -\frac {3 \left (1-i \sqrt {3}\right ) \cos \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}\right ){}^{2/3}}{-2 i 2^{2/3} \sqrt {3} \cos ^2(K[3])-2\ 2^{2/3} \cos ^2(K[3])+4 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}\right ){}^{2/3} \cos (K[3])+3 i \sqrt [3]{2} \sqrt {3} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}-3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}+i \sqrt [3]{2} \sqrt {3} \sqrt {9 c_1{}^2-4 \cos ^3(K[3])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}-\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[3])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}}dK[3]\& \right ]\left [\frac {t}{4}+c_2\right ]\right )}{2^{2/3} \sqrt [3]{81 c_1+\sqrt {6561 c_1{}^2-2916 \cos ^3\left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}\right ){}^{2/3}}{-2 i 2^{2/3} \sqrt {3} \cos ^2(K[3])-2\ 2^{2/3} \cos ^2(K[3])+4 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}\right ){}^{2/3} \cos (K[3])+3 i \sqrt [3]{2} \sqrt {3} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}-3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}+i \sqrt [3]{2} \sqrt {3} \sqrt {9 c_1{}^2-4 \cos ^3(K[3])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}-\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[3])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}}dK[3]\& \right ]\left [\frac {t}{4}+c_2\right ]\right )}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{81 c_1+\sqrt {6561 c_1{}^2-2916 \cos ^3\left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}\right ){}^{2/3}}{-2 i 2^{2/3} \sqrt {3} \cos ^2(K[3])-2\ 2^{2/3} \cos ^2(K[3])+4 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}\right ){}^{2/3} \cos (K[3])+3 i \sqrt [3]{2} \sqrt {3} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}-3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}+i \sqrt [3]{2} \sqrt {3} \sqrt {9 c_1{}^2-4 \cos ^3(K[3])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}-\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[3])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}}dK[3]\& \right ]\left [\frac {t}{4}+c_2\right ]\right )}}}{6 \sqrt [3]{2}},x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}\right ){}^{2/3}}{-2 i 2^{2/3} \sqrt {3} \cos ^2(K[3])-2\ 2^{2/3} \cos ^2(K[3])+4 \left (3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}\right ){}^{2/3} \cos (K[3])+3 i \sqrt [3]{2} \sqrt {3} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}-3 \sqrt [3]{2} c_1 \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}+i \sqrt [3]{2} \sqrt {3} \sqrt {9 c_1{}^2-4 \cos ^3(K[3])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}-\sqrt [3]{2} \sqrt {9 c_1{}^2-4 \cos ^3(K[3])} \sqrt [3]{3 c_1+\sqrt {9 c_1{}^2-4 \cos ^3(K[3])}}}dK[3]\& \right ]\left [\frac {t}{4}+c_2\right ]\right \}\right \}\] ✓ Maple : cpu = 1.252 (sec), leaf count = 108
\[\left \{\left [\left \{x \left (t \right ) = \RootOf \left (c_{2}+t -2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\cos \left (\textit {\_f} \right )-\sqrt {-\left (\cos ^{2}\left (\textit {\_f} \right )\right )-4 \cos \left (2 \textit {\_f} \right )-4}\, \tan \left (\RootOf \left (2 \textit {\_Z} \cos \left (\textit {\_f} \right )+3 c_{1} \sqrt {-\left (\cos ^{2}\left (\textit {\_f} \right )\right )}-3 \sqrt {-\left (\cos ^{2}\left (\textit {\_f} \right )\right )}\, \ln \left (\frac {9 \left (\cos ^{2}\left (\textit {\_f} \right )\right )}{4 \cos \left (\textit {\_Z} \right )^{2}}\right )\right )\right )}d \textit {\_f} \right )\right )\right \}, \left \{y \left (t \right ) = \sqrt {\cos \left (x \left (t \right )\right )+\frac {d}{d t}x \left (t \right )}, y \left (t \right ) = -\sqrt {\cos \left (x \left (t \right )\right )+\frac {d}{d t}x \left (t \right )}\right \}\right ]\right \}\]