\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-{\frac { \left ( -{a}^{2} \left ( \cos \left ( x \right ) \right ) ^{2}- \left ( 3-2\,a \right ) \cos \left ( x \right ) -3+3\,a \right ) y \left ( x \right ) }{ \left ( \sin \left ( x \right ) \right ) ^{2}}}=0} \]
Mathematica: cpu = 0.726092 (sec), leaf count = 232 \[ \left \{\left \{y(x)\to \frac {c_2 (1-\cos (x))^{\frac {1}{2}-a} \left (-\frac {(2 a-1) (\cos (x)+1)}{-2 a \cos (x)+\cos (x)+2}\right )^{a+\frac {1}{2}} (-2 a \cos (x)+\cos (x)+2) \left (\frac {(2 a-1) (\cos (x)-1)}{(2 a-1) \cos (x)-2}\right )^{a-\frac {1}{2}} (\cos (x)+1)^{-a-\frac {1}{2}} F_1\left (2 a;a-\frac {3}{2},a+\frac {1}{2};2 a+1;\frac {3-2 a}{-2 a \cos (x)+\cos (x)+2},\frac {2 a+1}{-2 a \cos (x)+\cos (x)+2}\right ) \exp \left (\frac {1}{2} (a-2) \log (1-\cos (x))+\frac {1}{2} a \log (\cos (x)+1)\right )}{2 (1-2 a)^2 a}+c_1 (-2 a \cos (x)+\cos (x)+2) \exp \left (\frac {1}{2} (a-2) \log (1-\cos (x))+\frac {1}{2} a \log (\cos (x)+1)\right )\right \}\right \} \]
Maple: cpu = 0.312 (sec), leaf count = 99 \[ \left \{ y \left ( x \right ) ={{\it \_C1}\, \left ( -2+ \left ( 2\,a-1 \right ) \cos \left ( x \right ) \right ) \sqrt [4]{2\,\cos \left ( x \right ) +2} \left ( \sin \left ( x \right ) \right ) ^{a-{\frac {1}{2}}} \left ( -2\,\cos \left ( x \right ) +2 \right ) ^{-{\frac {3}{4}}}}+{{ \it \_C2} {\mbox {$_2$F$_1$}(a-{\frac {1}{2}},-{\frac {1}{2}}-a;\,{\frac {3}{2}}-a;\,{\frac {\cos \left ( x \right ) }{2}}+{\frac {1}{2}})} \left ( -1+\cos \left ( x \right ) \right ) ^{{\frac {a}{2}}-{\frac {1}{ 4}}} \left ( \cos \left ( x \right ) +1 \right ) ^{-{\frac {1}{4}}-{\frac {a}{2}}} \left ( 2\,\cos \left ( x \right ) +2 \right ) ^{{\frac {3}{4}}} \left ( -2\,\cos \left ( x \right ) +2 \right ) ^{-{\frac {3}{4}}}} \right \} \]