\[ \boxed { \left ( b+a \left ( \sin \left ( y \left ( x \right ) \right ) \right ) ^{2} \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +a \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}\cos \left ( y \left ( x \right ) \right ) \sin \left ( y \left ( x \right ) \right ) +Ay \left ( x \right ) \left ( c+a \left ( \sin \left ( y \left ( x \right ) \right ) \right ) ^{2} \right ) =0} \]
Mathematica: cpu = 100.860808 (sec), leaf count = 48 \[ \text {DSolve}\left [A y(x) \left (a \sin ^2(y(x))+c\right )+y''(x) \left (a \sin ^2(y(x))+b\right )+a y'(x)^2 \sin (y(x)) \cos (y(x))=0,y(x),x\right ] \]
Maple: cpu = 0.265 (sec), leaf count = 146 \[ \left \{ \int ^{y \left ( x \right ) }\!{\sqrt {2} \left ( b+a \left ( \sin \left ( {\it \_a} \right ) \right ) ^{2} \right ) {\frac {1}{\sqrt { \left ( b+a \left ( \sin \left ( {\it \_a} \right ) \right ) ^{2} \right ) \left ( 2\,Aa{\it \_a}\,\cos \left ( {\it \_a} \right ) \sin \left ( {\it \_a} \right ) -Aa \left ( \sin \left ( {\it \_a} \right ) \right ) ^{2}-Aa{{\it \_a}}^{2}-2\,Ac{{\it \_a}}^{2}+2\,{\it \_C1} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\! -{\sqrt {2} \left ( b+a \left ( \sin \left ( {\it \_a} \right ) \right ) ^ {2} \right ) {\frac {1}{\sqrt { \left ( b+a \left ( \sin \left ( {\it \_a} \right ) \right ) ^{2} \right ) \left ( 2\,Aa{\it \_a}\,\cos \left ( { \it \_a} \right ) \sin \left ( {\it \_a} \right ) -Aa \left ( \sin \left ( {\it \_a} \right ) \right ) ^{2}-Aa{{\it \_a}}^{2}-2\,Ac{{\it \_a}}^{2} +2\,{\it \_C1} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \]