\[ A y(x) \left (a \sin ^2(y(x))+c\right )+y''(x) \left (a \sin ^2(y(x))+b\right )+a y'(x)^2 \sin (y(x)) \cos (y(x))=0 \] ✓ Mathematica : cpu = 25.2689 (sec), leaf count = 176
DSolve[A*(c + a*Sin[y[x]]^2)*y[x] + a*Cos[y[x]]*Sin[y[x]]*Derivative[1][y][x]^2 + (b + a*Sin[y[x]]^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} \sqrt {\cos (2 K[1]) a-a-2 b}}{\sqrt {2 a A K[1]^2+4 A c K[1]^2-2 a A \sin (2 K[1]) K[1]+2 c_1-a A \cos (2 K[1])}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} \sqrt {\cos (2 K[2]) a-a-2 b}}{\sqrt {2 a A K[2]^2+4 A c K[2]^2-2 a A \sin (2 K[2]) K[2]+2 c_1-a A \cos (2 K[2])}}dK[2]\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 0.45 (sec), leaf count = 133
dsolve((b+a*sin(y(x))^2)*diff(diff(y(x),x),x)+a*diff(y(x),x)^2*cos(y(x))*sin(y(x))+A*y(x)*(c+a*sin(y(x))^2)=0,y(x))
\[\int _{}^{y \left (x \right )}-\frac {2 \left (b +a \left (\sin ^{2}\left (\textit {\_a} \right )\right )\right )}{\sqrt {-2 \left (b +a \left (\sin ^{2}\left (\textit {\_a} \right )\right )\right ) \left (A a \left (\sin ^{2}\left (\textit {\_a} \right )\right )-2 A a \textit {\_a} \sin \left (\textit {\_a} \right ) \cos \left (\textit {\_a} \right )+\textit {\_a}^{2} \left (a +2 c \right ) A -2 c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0\]