\[ \boxed { {\it d4y} \left ( x \right ) \left ( \sin \left ( x \right ) \right ) ^{4}+2\, \left ( {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) \right ) \left ( \sin \left ( x \right ) \right ) ^{3}\cos \left ( x \right ) + \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) \left ( \sin \left ( x \right ) \right ) ^{2} \left ( \left ( \sin \left ( x \right ) \right ) ^{2}-3 \right ) + \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) \sin \left ( x \right ) \cos \left ( x \right ) \left ( 2\, \left ( \sin \left ( x \right ) \right ) ^{2}+3 \right ) + \left ( {a}^{4} \left ( \sin \left ( x \right ) \right ) ^{4}-3 \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.206526 (sec), leaf count = 73 \[ \text {DSolve}\left [y(x) \left (a^4 \sin ^4(x)-3\right )+y^{(4)}(x) \sin ^4(x)+2 y^{(3)}(x) \sin ^3(x) \cos (x)+\left (\sin ^2(x)-3\right ) \sin ^2(x) y''(x)+\left (2 \sin ^2(x)+3\right ) \sin (x) \cos (x) y'(x)=0,y(x),x\right ] \]
Maple: cpu = 0.639 (sec), leaf count = 345 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sin \left ( x \right ) {\mbox {$_2$F$_1$}({\frac {3}{4}}+{\frac {1}{4}\sqrt {-i \left ( -4\,\sqrt {a+i}\sqrt {a-i}\sqrt {a-1}\sqrt {a+1}+5\,i \right ) }},{\frac {3}{4}}-{\frac {1}{4}\sqrt {-i \left ( -4\,\sqrt {a+i}\sqrt {a-i}\sqrt {a-1}\sqrt {a+1}+5\,i \right ) }};\,{\frac {1}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})} +{\it \_C2}\,\sin \left ( x \right ) {\mbox {$_2$F$_1$}({\frac {3}{4}}+{\frac {1}{4}\sqrt {-i \left ( 4\,\sqrt {a+i}\sqrt {a-i}\sqrt {a-1}\sqrt {a+1}+5\,i \right ) }},{\frac {3}{4}}-{\frac {1}{4}\sqrt {-i \left ( 4\,\sqrt {a+i}\sqrt {a-i}\sqrt {a-1}\sqrt {a+1}+5\,i \right ) }};\,{\frac {1}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})} +{\it \_C3}\,\sin \left ( x \right ) \cos \left ( x \right ) {\mbox {$_2$F$_1$}({\frac {5}{4}}+{\frac {1}{4}\sqrt {-i \left ( -4\,\sqrt {a+i}\sqrt {a-i}\sqrt {a-1}\sqrt {a+1}+5\,i \right ) }},{\frac {5}{4}}-{\frac {1}{4}\sqrt {-i \left ( -4\,\sqrt {a+i}\sqrt {a-i}\sqrt {a-1}\sqrt {a+1}+5\,i \right ) }};\,{\frac {3}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})} +{\it \_C4}\,\sin \left ( x \right ) \cos \left ( x \right ) {\mbox {$_2$F$_1$}({\frac {5}{4}}+{\frac {1}{4}\sqrt {-i \left ( 4\,\sqrt {a+i}\sqrt {a-i}\sqrt {a-1}\sqrt {a+1}+5\,i \right ) }},{\frac {5}{4}}-{\frac {1}{4}\sqrt {-i \left ( 4\,\sqrt {a+i}\sqrt {a-i}\sqrt {a-1}\sqrt {a+1}+5\,i \right ) }};\,{\frac {3}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})} \right \} \]