\[ \boxed { \left ( {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) \right ) \left ( \sin \left ( x \right ) \right ) ^{2}+3\, \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) \sin \left ( x \right ) \cos \left ( x \right ) + \left ( \cos \left ( 2\,x \right ) +4\,\nu \, \left ( \nu +1 \right ) \left ( \sin \left ( x \right ) \right ) ^{2} \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +2\,\nu \, \left ( \nu +1 \right ) y \left ( x \right ) \sin \left ( 2\,x \right ) =0} \]
Mathematica: cpu = 0.112514 (sec), leaf count = 58 \[ \text {DSolve}\left [y'(x) \left (4 \nu (\nu +1) \sin ^2(x)+\cos (2 x)\right )+2 \nu (\nu +1) y(x) \sin (2 x)+y^{(3)}(x) \sin ^2(x)+3 \sin (x) \cos (x) y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.187 (sec), leaf count = 113 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( {\mbox {$_2$F$_1$}(-{\frac {\nu }{2}},{\frac {\nu }{2}}+{\frac {1}{2}};\,{\frac {1}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})} \right ) ^{2}+{\it \_C2}\, \left ( \cos \left ( 2\,x \right ) +1 \right ) \left ( {\mbox {$_2$F$_1$}(1+{\frac {\nu }{2}},{\frac {1}{2}}-{\frac {\nu }{2}};\,{\frac {3}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})} \right ) ^{2}+{\it \_C3}\, {\mbox {$_2$F$_1$}(-{\frac {\nu }{2}},{\frac {\nu }{2}}+{\frac {1}{2}};\,{\frac {1}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})} \sqrt {\cos \left ( 2\,x \right ) +1} {\mbox {$_2$F$_1$}(1+{\frac {\nu }{2}},{\frac {1}{2}}-{\frac {\nu }{2}};\,{\frac {3}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})} \right \} \]