\[ y''(x)=y(x) \csc ^2(x) \left (-\left (-a^2 \cos ^2(x)-(3-2 a) \cos (x)+3 a-3\right )\right ) \] ✓ Mathematica : cpu = 0.741325 (sec), leaf count = 194
\[\left \{\left \{y(x)\to \frac {c_1 \sin ^2(x)^{a/2} (-2 a \cos (x)+\cos (x)+2)}{1-\cos (x)}-\frac {c_2 \sin ^2(x)^{-a} (1-\cos (x))^{\frac {a-1}{2}} (\cos (x)+1)^{\frac {a+1}{2}} \left (\frac {(2 a-1) (\cos (x)-1)}{(2 a-1) \cos (x)-2}\right )^{a-\frac {1}{2}} \left (\frac {(2 a-1) (\cos (x)+1)}{(2 a-1) \cos (x)-2}\right )^{a-\frac {1}{2}} F_1\left (2 a;a-\frac {3}{2},a+\frac {1}{2};2 a+1;\frac {3-2 a}{-2 a \cos (x)+\cos (x)+2},\frac {2 a+1}{-2 a \cos (x)+\cos (x)+2}\right )}{4 a^2-2 a}\right \}\right \}\]
✓ Maple : cpu = 0.47 (sec), leaf count = 91
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{\mbox {$_2$F$_1$}(a-{\frac {1}{2}},-{\frac {1}{2}}-a;\,{\frac {3}{2}}-a;\,{\frac {\cos \left ( x \right ) }{2}}+{\frac {1}{2}})} \left ( \cos \left ( x \right ) +1 \right ) ^{-{\frac {1}{4}}-{\frac {a}{2}}}\sqrt {2\,\cos \left ( x \right ) +2} \left ( \cos \left ( x \right ) -1 \right ) ^{{\frac {a}{2}}-{\frac {1}{4}}}+2\, \left ( \sin \left ( x \right ) \right ) ^{a-1/2} \left ( -1+ \left ( a-1/2 \right ) \cos \left ( x \right ) \right ) {\it \_C1} \right ) \sqrt [4]{2\,\cos \left ( x \right ) +2} \left ( -2\,\cos \left ( x \right ) +2 \right ) ^{-{\frac {3}{4}}}} \right \} \]