\[ y''(x)=y(x) \csc ^2(x) \left (-\left (-a^2 \cos ^2(x)-(3-2 a) \cos (x)+3 a-3\right )\right ) \] ✓ Mathematica : cpu = 0.658445 (sec), leaf count = 236
\[\left \{\left \{y(x)\to \frac {c_2 \sqrt {1-\cos (x)} \left (-\frac {(2 a-1) (\cos (x)+1)}{-2 a \cos (x)+\cos (x)+2}\right )^{a+\frac {1}{2}} (-2 a \cos (x)+\cos (x)+2) \left (1-\cos ^2(x)\right )^{-a} \left (\frac {(2 a-1) (\cos (x)-1)}{(2 a-1) \cos (x)-2}\right )^{a-\frac {1}{2}} F_1\left (2 a;a-\frac {3}{2},a+\frac {1}{2};2 a+1;\frac {3-2 a}{-2 a \cos (x)+\cos (x)+2},\frac {2 a+1}{-2 a \cos (x)+\cos (x)+2}\right ) \exp \left (\frac {1}{2} (a-2) \log (1-\cos (x))+\frac {1}{2} a \log (\cos (x)+1)\right )}{2 (1-2 a)^2 a \sqrt {\cos (x)+1}}+c_1 (-2 a \cos (x)+\cos (x)+2) \exp \left (\frac {1}{2} (a-2) \log (1-\cos (x))+\frac {1}{2} a \log (\cos (x)+1)\right )\right \}\right \}\] ✓ Maple : cpu = 2.643 (sec), leaf count = 91
\[ \left \{ y \left ( x \right ) ={\sqrt [4]{2\,\cos \left ( x \right ) +2} \left ( {\it \_C2}\,{\mbox {$_2$F$_1$}(a-{\frac {1}{2}},-{\frac {1}{2}}-a;\,{\frac {3}{2}}-a;\,{\frac {\cos \left ( x \right ) }{2}}+{\frac {1}{2}})} \left ( \cos \left ( x \right ) +1 \right ) ^{-{\frac {1}{4}}-{\frac {a}{2}}}\sqrt {2\,\cos \left ( x \right ) +2} \left ( -1+\cos \left ( x \right ) \right ) ^{{\frac {a}{2}}-{\frac {1}{4}}}+2\,{\it \_C1}\, \left ( -1+ \left ( a-1/2 \right ) \cos \left ( x \right ) \right ) \left ( \sin \left ( x \right ) \right ) ^{a-1/2} \right ) \left ( -2\,\cos \left ( x \right ) +2 \right ) ^{-{\frac {3}{4}}}} \right \} \]