\[ y'(x) \cos (a y(x))-b (1-c \cos (a y(x))) \sqrt {c \cos (a y(x))+\cos ^2(a y(x))-1}=0 \] ✓ Mathematica : cpu = 50.9914 (sec), leaf count = 6218
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {2} (\cos (a \text {$\#$1})+1) \sqrt {\frac {2 c \cos (a \text {$\#$1})+\cos (2 a \text {$\#$1})-1}{(\cos (a \text {$\#$1})+1)^2}} \left (-\frac {\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\left (\frac {\sqrt {c-1}}{\sqrt {c+1}}+\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}\right )-2 \sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \Pi \left (\frac {\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\frac {\sqrt {c-1}}{\sqrt {c+1}}\right ) \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (-\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}\right )\right ) \sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )^2}{\sqrt {c-1} \sqrt {c+1} \sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (-\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \sqrt {-c \tan ^4\left (\frac {a \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {a \text {$\#$1}}{2}\right )+c}}+\frac {\sqrt {c-1} \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\left (\frac {\sqrt {c-1}}{\sqrt {c+1}}+\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}\right )-2 \sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \Pi \left (\frac {\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\frac {\sqrt {c-1}}{\sqrt {c+1}}\right ) \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (-\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}\right )\right ) \sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )^2}{(c+1)^{3/2} \sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (-\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \sqrt {-c \tan ^4\left (\frac {a \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {a \text {$\#$1}}{2}\right )+c}}+\frac {\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\frac {\sqrt {c-1}}{\sqrt {c+1}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}\right )-2 \sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \Pi \left (\frac {\left (\frac {\sqrt {c-1}}{\sqrt {c+1}}+\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}\right )\right ) \sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )^2}{\sqrt {c-1} \sqrt {c+1} \sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (-\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \sqrt {-c \tan ^4\left (\frac {a \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {a \text {$\#$1}}{2}\right )+c}}-\frac {\sqrt {c-1} \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\frac {\sqrt {c-1}}{\sqrt {c+1}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}\right )-2 \sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \Pi \left (\frac {\left (\frac {\sqrt {c-1}}{\sqrt {c+1}}+\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right )^2}\right )\right ) \sqrt {\frac {\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}\right )}{\left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}+\sqrt {\frac {\sqrt {c^2+4}-2}{c}}\right ) \left (\sqrt {\frac {-\sqrt {c^2+4}-2}{c}}-\tan \left (\frac {a \text {$\#$1}}{2}\right )\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )+\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}{\left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )}} \left (\tan \left (\frac {a \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right )^2}{(c+1)^{3/2} \sqrt {\frac {-\sqrt {c^2+4}-2}{c}} \left (-\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\frac {\sqrt {c-1}}{\sqrt {c+1}}-\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \left (\sqrt {-\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}-\sqrt {\frac {\sqrt {c^2+4}}{c}-\frac {2}{c}}\right ) \sqrt {-c \tan ^4\left (\frac {a \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {a \text {$\#$1}}{2}\right )+c}}-\frac {i F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{\sqrt {c^2+4}-2}} \tan \left (\frac {a \text {$\#$1}}{2}\right )\right )|\frac {\sqrt {c^2+4}-2}{-\sqrt {c^2+4}-2}\right ) \sqrt {1-\frac {c \tan ^2\left (\frac {a \text {$\#$1}}{2}\right )}{-\sqrt {c^2+4}-2}} \sqrt {1-\frac {c \tan ^2\left (\frac {a \text {$\#$1}}{2}\right )}{\sqrt {c^2+4}-2}}}{(c+1) \sqrt {-\frac {c}{\sqrt {c^2+4}-2}} \sqrt {-c \tan ^4\left (\frac {a \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {a \text {$\#$1}}{2}\right )+c}}\right )}{a \sqrt {2 c \cos (a \text {$\#$1})+\cos (2 a \text {$\#$1})-1}}\& \right ]\left [c_1-\frac {b x}{\sqrt {2}}\right ]\right \}\right \}\]
✓ Maple : cpu = 0.2 (sec), leaf count = 48
\[ \left \{ x+\int ^{y \left ( x \right ) }\!2\,{\frac {\cos \left ( {\it \_a}\,a \right ) }{b \left ( c\cos \left ( {\it \_a}\,a \right ) -1 \right ) \sqrt {2\,\cos \left ( 2\,{\it \_a}\,a \right ) -2+4\,c\cos \left ( {\it \_a}\,a \right ) }}}{d{\it \_a}}+{\it \_C1}=0 \right \} \]