#### 2.81   ODE No. 81

$y'(x)+2 \tan (x) \tan (y(x))-1=0$ Mathematica : cpu = 1.33772 (sec), leaf count = 220

$\text {Solve}\left [c_1=\frac {\frac {1}{2} \left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right ) \sqrt [4]{1-\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};\left (i \cot (x)+\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}\right )^2\right )+i \tan (x)}{\sqrt [4]{-1+\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2}},y(x)\right ]$ Maple : cpu = 1.052 (sec), leaf count = 78

$\left \{ {\it \_C1}+{\tan \left ( x \right ) {\frac {1}{\sqrt [4]{{\frac { \left ( 1+ \left ( \tan \left ( y \left ( x \right ) \right ) \right ) ^{2} \right ) \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }{ \left ( \tan \left ( y \left ( x \right ) \right ) \tan \left ( x \right ) -1 \right ) ^{2}}}}}}}+{\frac {\tan \left ( y \left ( x \right ) \right ) +\tan \left ( x \right ) }{2\,\tan \left ( y \left ( x \right ) \right ) \tan \left ( x \right ) -2}{\mbox {_2F_1}({\frac {1}{2}},{\frac {5}{4}};\,{\frac {3}{2}};\,-{\frac { \left ( \tan \left ( y \left ( x \right ) \right ) +\tan \left ( x \right ) \right ) ^{2}}{ \left ( \tan \left ( y \left ( x \right ) \right ) \tan \left ( x \right ) -1 \right ) ^{2}}})}}=0 \right \}$