#### 2.514   ODE No. 514

$y'(x)^2 (a \cos (y(x))+b)-c \cos (y(x))+d=0$ Mathematica : cpu = 9.97534 (sec), leaf count = 605

$\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {4 \sin ^2\left (\frac {\text {\#1}}{2}\right ) \csc (\text {\#1}) \sqrt {a \cos (\text {\#1})+b} \sqrt {\frac {\cot ^2\left (\frac {\text {\#1}}{2}\right ) (c-d)}{c+d}} \sqrt {\frac {\csc ^2\left (\frac {\text {\#1}}{2}\right ) (a+b) (d-c \cos (\text {\#1}))}{a d+b c}} \left (c (a+b) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {\#1})) \csc ^2\left (\frac {\text {\#1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right )|\frac {2 (b c+a d)}{(a+b) (c+d)}\right )+a (d-c) \Pi \left (\frac {b c+a d}{a c+b c};\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {\#1})) \csc ^2\left (\frac {\text {\#1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right )|\frac {2 (b c+a d)}{(a+b) (c+d)}\right )\right )}{c (a+b) \sqrt {c \cos (\text {\#1})-d} \sqrt {\frac {\csc ^2\left (\frac {\text {\#1}}{2}\right ) (c-d) (a \cos (\text {\#1})+b)}{a d+b c}}}\& \right ][c_1-x]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {4 \sin ^2\left (\frac {\text {\#1}}{2}\right ) \csc (\text {\#1}) \sqrt {a \cos (\text {\#1})+b} \sqrt {\frac {\cot ^2\left (\frac {\text {\#1}}{2}\right ) (c-d)}{c+d}} \sqrt {\frac {\csc ^2\left (\frac {\text {\#1}}{2}\right ) (a+b) (d-c \cos (\text {\#1}))}{a d+b c}} \left (c (a+b) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {\#1})) \csc ^2\left (\frac {\text {\#1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right )|\frac {2 (b c+a d)}{(a+b) (c+d)}\right )+a (d-c) \Pi \left (\frac {b c+a d}{a c+b c};\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {\#1})) \csc ^2\left (\frac {\text {\#1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right )|\frac {2 (b c+a d)}{(a+b) (c+d)}\right )\right )}{c (a+b) \sqrt {c \cos (\text {\#1})-d} \sqrt {\frac {\csc ^2\left (\frac {\text {\#1}}{2}\right ) (c-d) (a \cos (\text {\#1})+b)}{a d+b c}}}\& \right ][c_1+x]\right \}\right \}$ Maple : cpu = 0.257 (sec), leaf count = 87

$\left \{ x-\int ^{y \left ( x \right ) }\!{(a\cos \left ( {\it \_a} \right ) +b){\frac {1}{\sqrt { \left ( a\cos \left ( {\it \_a} \right ) +b \right ) \left ( c\cos \left ( {\it \_a} \right ) -d \right ) }}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!-{(a\cos \left ( {\it \_a} \right ) +b){\frac {1}{\sqrt { \left ( a\cos \left ( {\it \_a} \right ) +b \right ) \left ( c\cos \left ( {\it \_a} \right ) -d \right ) }}}}{d{\it \_a}}-{\it \_C1}=0,y \left ( x \right ) =\arccos \left ( {\frac {d}{c}} \right ) \right \}$