\[ \text {Global$\grave { }$y}'(\text {Global$\grave { }$x}) \cos (\text {Global$\grave { }$a} \text {Global$\grave { }$y}(\text {Global$\grave { }$x}))-\text {Global$\grave { }$b} (1-\text {Global$\grave { }$c} \cos (\text {Global$\grave { }$a} \text {Global$\grave { }$y}(\text {Global$\grave { }$x}))) \sqrt {\text {Global$\grave { }$c} \cos (\text {Global$\grave { }$a} \text {Global$\grave { }$y}(\text {Global$\grave { }$x}))+\cos ^2(\text {Global$\grave { }$a} \text {Global$\grave { }$y}(\text {Global$\grave { }$x}))-1}=0 \] ✓ Mathematica : cpu = 52.3987 (sec), leaf count = 6218
\[\left \{\left \{\text {Global$\grave { }$y}(\text {Global$\grave { }$x})\to \text {InverseFunction}\left [\frac {\sqrt {2} (\cos (\text {Global$\grave { }$a} \text {$\#$1})+1) \sqrt {\frac {2 \text {Global$\grave { }$c} \cos (\text {Global$\grave { }$a} \text {$\#$1})+\cos (2 \text {Global$\grave { }$a} \text {$\#$1})-1}{(\cos (\text {Global$\grave { }$a} \text {$\#$1})+1)^2}} \left (-\frac {\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}+\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}\right )-2 \sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \Pi \left (\frac {\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}\right ) \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}\right )\right ) \sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )^2}{\sqrt {\text {Global$\grave { }$c}-1} \sqrt {\text {Global$\grave { }$c}+1} \sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \sqrt {-\text {Global$\grave { }$c} \tan ^4\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\text {Global$\grave { }$c}}}+\frac {\sqrt {\text {Global$\grave { }$c}-1} \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}+\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}\right )-2 \sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \Pi \left (\frac {\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}\right ) \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}\right )\right ) \sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )^2}{(\text {Global$\grave { }$c}+1)^{3/2} \sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \sqrt {-\text {Global$\grave { }$c} \tan ^4\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\text {Global$\grave { }$c}}}+\frac {\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}\right )-2 \sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \Pi \left (\frac {\left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}+\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}\right )\right ) \sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )^2}{\sqrt {\text {Global$\grave { }$c}-1} \sqrt {\text {Global$\grave { }$c}+1} \sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \sqrt {-\text {Global$\grave { }$c} \tan ^4\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\text {Global$\grave { }$c}}}-\frac {\sqrt {\text {Global$\grave { }$c}-1} \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}\right )-2 \sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \Pi \left (\frac {\left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}+\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}}\right )|\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )^2}\right )\right ) \sqrt {\frac {\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}}-\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}} \sqrt {\frac {\sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}{\left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )}} \left (\tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right )^2}{(\text {Global$\grave { }$c}+1)^{3/2} \sqrt {\frac {-\sqrt {\text {Global$\grave { }$c}^2+4}-2}{\text {Global$\grave { }$c}}} \left (-\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\frac {\sqrt {\text {Global$\grave { }$c}-1}}{\sqrt {\text {Global$\grave { }$c}+1}}-\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \left (\sqrt {-\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}-\sqrt {\frac {\sqrt {\text {Global$\grave { }$c}^2+4}}{\text {Global$\grave { }$c}}-\frac {2}{\text {Global$\grave { }$c}}}\right ) \sqrt {-\text {Global$\grave { }$c} \tan ^4\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\text {Global$\grave { }$c}}}-\frac {i F\left (i \sinh ^{-1}\left (\sqrt {-\frac {\text {Global$\grave { }$c}}{\sqrt {\text {Global$\grave { }$c}^2+4}-2}} \tan \left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )\right )|\frac {\sqrt {\text {Global$\grave { }$c}^2+4}-2}{-\sqrt {\text {Global$\grave { }$c}^2+4}-2}\right ) \sqrt {1-\frac {\text {Global$\grave { }$c} \tan ^2\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )}{-\sqrt {\text {Global$\grave { }$c}^2+4}-2}} \sqrt {1-\frac {\text {Global$\grave { }$c} \tan ^2\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )}{\sqrt {\text {Global$\grave { }$c}^2+4}-2}}}{(\text {Global$\grave { }$c}+1) \sqrt {-\frac {\text {Global$\grave { }$c}}{\sqrt {\text {Global$\grave { }$c}^2+4}-2}} \sqrt {-\text {Global$\grave { }$c} \tan ^4\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )-4 \tan ^2\left (\frac {\text {Global$\grave { }$a} \text {$\#$1}}{2}\right )+\text {Global$\grave { }$c}}}\right )}{\text {Global$\grave { }$a} \sqrt {2 \text {Global$\grave { }$c} \cos (\text {Global$\grave { }$a} \text {$\#$1})+\cos (2 \text {Global$\grave { }$a} \text {$\#$1})-1}}\& \right ]\left [c_1-\frac {\text {Global$\grave { }$b} \text {Global$\grave { }$x}}{\sqrt {2}}\right ]\right \}\right \}\]
✓ Maple : cpu = 0.206 (sec), leaf count = 48
\[ \left \{ x+\int ^{y \left ( x \right ) }\!2\,{\frac {\cos \left ( a{\it \_a} \right ) }{b \left ( c\cos \left ( a{\it \_a} \right ) -1 \right ) \sqrt {2\,\cos \left ( 2\,a{\it \_a} \right ) -2+4\,c\cos \left ( a{\it \_a} \right ) }}}{d{\it \_a}}+{\it \_C1}=0 \right \} \]