\[ y''(x)=y(x) \csc ^2(x) \left (a^2 \cos ^2(x)+(3-2 a) \cos (x)-3 a+3\right ) \] ✓ Mathematica : cpu = 0.752464 (sec), leaf count = 236
DSolve[Derivative[2][y][x] == (3 - 3*a + (3 - 2*a)*Cos[x] + a^2*Cos[x]^2)*Csc[x]^2*y[x],y[x],x]
\[\left \{\left \{y(x)\to \frac {c_2 \sqrt {1-\cos (x)} \left (-\frac {(2 a-1) (\cos (x)+1)}{-2 a \cos (x)+\cos (x)+2}\right )^{a+\frac {1}{2}} (-2 a \cos (x)+\cos (x)+2) \left (1-\cos ^2(x)\right )^{-a} \left (\frac {(2 a-1) (\cos (x)-1)}{(2 a-1) \cos (x)-2}\right )^{a-\frac {1}{2}} F_1\left (2 a;a-\frac {3}{2},a+\frac {1}{2};2 a+1;\frac {3-2 a}{-2 a \cos (x)+\cos (x)+2},\frac {2 a+1}{-2 a \cos (x)+\cos (x)+2}\right ) \exp \left (\frac {1}{2} (a-2) \log (1-\cos (x))+\frac {1}{2} a \log (\cos (x)+1)\right )}{2 (1-2 a)^2 a \sqrt {\cos (x)+1}}+c_1 (-2 a \cos (x)+\cos (x)+2) \exp \left (\frac {1}{2} (a-2) \log (1-\cos (x))+\frac {1}{2} a \log (\cos (x)+1)\right )\right \}\right \}\] ✓ Maple : cpu = 0.415 (sec), leaf count = 91
dsolve(diff(diff(y(x),x),x) = -(-a^2*cos(x)^2-(3-2*a)*cos(x)-3+3*a)/sin(x)^2*y(x),y(x))
\[y \left (x \right ) = \frac {\left (2 \cos \left (x \right )+2\right )^{\frac {1}{4}} \left (c_{2} \hypergeom \left (\left [a -\frac {1}{2}, -\frac {1}{2}-a \right ], \left [\frac {3}{2}-a \right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \left (\cos \left (x \right )+1\right )^{-\frac {1}{4}-\frac {a}{2}} \sqrt {2 \cos \left (x \right )+2}\, \left (\cos \left (x \right )-1\right )^{\frac {a}{2}-\frac {1}{4}}+2 \left (-1+\left (a -\frac {1}{2}\right ) \cos \left (x \right )\right ) c_{1} \left (\sin ^{a -\frac {1}{2}}\left (x \right )\right )\right )}{\left (-2 \cos \left (x \right )+2\right )^{\frac {3}{4}}}\]