\[ y''(x)=y(x) \left (-\csc ^2(x)\right ) \left (a \cos ^2(x)+b \sin ^2(x)+c\right ) \] ✓ Mathematica : cpu = 0.34127 (sec), leaf count = 104
DSolve[Derivative[2][y][x] == -(Csc[x]^2*(c + a*Cos[x]^2 + b*Sin[x]^2)*y[x]),y[x],x]
\[\left \{\left \{y(x)\to c_1 \sqrt [4]{\cos ^2(x)-1} P_{\frac {1}{2} \left (2 \sqrt {b-a}-1\right )}^{\frac {1}{2} \sqrt {-4 a-4 c+1}}(\cos (x))+c_2 \sqrt [4]{\cos ^2(x)-1} Q_{\frac {1}{2} \left (2 \sqrt {b-a}-1\right )}^{\frac {1}{2} \sqrt {-4 a-4 c+1}}(\cos (x))\right \}\right \}\] ✓ Maple : cpu = 0.284 (sec), leaf count = 183
dsolve(diff(diff(y(x),x),x) = -(a*cos(x)^2+b*sin(x)^2+c)/sin(x)^2*y(x),y(x))
\[y \left (x \right ) = \frac {\sqrt {-2 \cos \left (2 x \right )+2}\, \left (2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {-4 a +1-4 c}}{4}} \left (\hypergeom \left (\left [\frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {3}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {2 \cos \left (2 x \right )+2}\, c_{2}+\hypergeom \left (\left [\frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {1}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1}\right )}{\sqrt {\sin \left (2 x \right )}}\]