\[ y'(x) \left (4 \nu (\nu +1) \sin ^2(x)+\cos (2 x)\right )+2 \nu (\nu +1) y(x) \sin (2 x)+y^{(3)}(x) \sin ^2(x)+3 \sin (x) \cos (x) y''(x)=0 \] ✓ Mathematica : cpu = 0.0731833 (sec), leaf count = 35
DSolve[2*nu*(1 + nu)*Sin[2*x]*y[x] + (Cos[2*x] + 4*nu*(1 + nu)*Sin[x]^2)*Derivative[1][y][x] + 3*Cos[x]*Sin[x]*Derivative[2][y][x] + Sin[x]^2*Derivative[3][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_3 P_{\nu }(\cos (x)) Q_{\nu }(\cos (x))+c_1 P_{\nu }(\cos (x)){}^2+c_2 Q_{\nu }(\cos (x)){}^2\right \}\right \}\] ✓ Maple : cpu = 0.216 (sec), leaf count = 113
dsolve(diff(diff(diff(y(x),x),x),x)*sin(x)^2+3*diff(diff(y(x),x),x)*sin(x)*cos(x)+(cos(2*x)+4*nu*(nu+1)*sin(x)^2)*diff(y(x),x)+2*nu*(nu+1)*y(x)*sin(2*x)=0,y(x))
\[y \left (x \right ) = c_{1} \hypergeom \left (\left [-\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{2}+c_{2} \left (\cos \left (2 x \right )+1\right ) \hypergeom \left (\left [1+\frac {\nu }{2}, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{2}+c_{3} \hypergeom \left (\left [-\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {\cos \left (2 x \right )+1}\, \hypergeom \left (\left [1+\frac {\nu }{2}, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\]