\[ \boxed { \left ( \sin \left ( x \right ) \right ) ^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( a \left ( \sin \left ( x \right ) \right ) ^{2}+n \left ( n-1 \right ) \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.171022 (sec), leaf count = 90 \[ \left \{\left \{y(x)\to c_1 \sqrt [4]{\cos ^2(x)-1} P_{\frac {1}{2} i \left (2 \sqrt {a}+i\right )}^{\frac {1}{2} (2 n-1)}(\cos (x))+c_2 \sqrt [4]{\cos ^2(x)-1} Q_{\frac {1}{2} i \left (2 \sqrt {a}+i\right )}^{\frac {1}{2} (2 n-1)}(\cos (x))\right \}\right \} \]
Maple: cpu = 0.203 (sec), leaf count = 125 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( {\frac {\cos \left ( 2 \,x \right ) }{2}}-{\frac {1}{2}} \right ) ^{{\frac {n}{2}}} {\mbox {$_2$F$_1$}({\frac {n}{2}}+{\frac {i}{2}}\sqrt {a},{\frac {n}{2}}-{\frac {i}{2}}\sqrt {a};\,{\frac {1}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})} +{{\it \_C2} \left ( 2\,\cos \left ( 2\,x \right ) +2 \right ) ^{{\frac {3 }{4}}} {\mbox {$_2$F$_1$}({\frac {1}{2}}+{\frac {i}{2}}\sqrt {a}+{\frac {n}{2}},{\frac {1}{2}}-{\frac {i}{2}}\sqrt {a}+{\frac {n}{2}};\,{\frac {3}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})} \left ( {\frac {\cos \left ( 2\,x \right ) }{2}}-{\frac {1}{2}} \right ) ^{{\frac {n}{2}}}\sqrt [4]{-2\,\cos \left ( 2\,x \right ) +2}{\frac {1}{ \sqrt {\sin \left ( 2\,x \right ) }}}} \right \} \]