\[ -f(x)+y^{(4)}(x) \sin ^6(x)+4 y^{(3)}(x) \sin ^5(x) \cos (x)-6 \sin ^6(x) y''(x)-4 \sin ^5(x) \cos (x) y'(x)+y(x) \sin ^6(x)=0 \] ✓ Mathematica : cpu = 7.53617 (sec), leaf count = 138
DSolve[-f[x] + Sin[x]^6*y[x] - 4*Cos[x]*Sin[x]^5*Derivative[1][y][x] - 6*Sin[x]^6*Derivative[2][y][x] + 4*Cos[x]*Sin[x]^5*Derivative[3][y][x] + Sin[x]^6*Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to x^3 \csc (x) \int _1^x\frac {1}{6} \csc ^5(K[4]) f(K[4])dK[4]+x^2 \csc (x) \int _1^x-\frac {1}{2} \csc ^5(K[3]) f(K[3]) K[3]dK[3]+x \csc (x) \int _1^x\frac {1}{2} \csc ^5(K[2]) f(K[2]) K[2]^2dK[2]+\csc (x) \int _1^x-\frac {1}{6} \csc ^5(K[1]) f(K[1]) K[1]^3dK[1]+c_4 x^3 \csc (x)+c_3 x^2 \csc (x)+c_2 x \csc (x)+c_1 \csc (x)\right \}\right \}\] ✓ Maple : cpu = 0.469 (sec), leaf count = 638
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)*sin(x)^6+4*diff(diff(diff(y(x),x),x),x)*sin(x)^5*cos(x)-6*diff(diff(y(x),x),x)*sin(x)^6-4*diff(y(x),x)*sin(x)^5*cos(x)+y(x)*sin(x)^6-f=0,y(x))
\[y \left (x \right ) = \frac {12 \left (x^{2}+\frac {20}{3}\right ) x \left ({\mathrm e}^{2 i x}-\frac {3 \,{\mathrm e}^{4 i x}}{2}+{\mathrm e}^{6 i x}-\frac {{\mathrm e}^{8 i x}}{4}-\frac {1}{4}\right ) f \ln \left (1-{\mathrm e}^{i x}\right )-80 i \left ({\mathrm e}^{2 i x}-\frac {3 \,{\mathrm e}^{4 i x}}{2}+{\mathrm e}^{6 i x}-\frac {{\mathrm e}^{8 i x}}{4}-\frac {1}{4}\right ) f \polylog \left (2, {\mathrm e}^{i x}\right )+72 i \left ({\mathrm e}^{2 i x}-\frac {3 \,{\mathrm e}^{4 i x}}{2}+{\mathrm e}^{6 i x}-\frac {{\mathrm e}^{8 i x}}{4}-\frac {1}{4}\right ) f \polylog \left (4, {\mathrm e}^{i x}\right )-12 x^{3} \left ({\mathrm e}^{2 i x}-\frac {3 \,{\mathrm e}^{4 i x}}{2}+{\mathrm e}^{6 i x}-\frac {{\mathrm e}^{8 i x}}{4}-\frac {1}{4}\right ) f \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-12 \left (x^{2}+\frac {20}{3}\right ) x \left ({\mathrm e}^{2 i x}-\frac {3 \,{\mathrm e}^{4 i x}}{2}+{\mathrm e}^{6 i x}-\frac {{\mathrm e}^{8 i x}}{4}-\frac {1}{4}\right ) f \ln \left (1+{\mathrm e}^{i x}\right )+80 i \left ({\mathrm e}^{2 i x}-\frac {3 \,{\mathrm e}^{4 i x}}{2}+{\mathrm e}^{6 i x}-\frac {{\mathrm e}^{8 i x}}{4}-\frac {1}{4}\right ) f \polylog \left (2, -{\mathrm e}^{i x}\right )-72 i \left ({\mathrm e}^{2 i x}-\frac {3 \,{\mathrm e}^{4 i x}}{2}+{\mathrm e}^{6 i x}-\frac {{\mathrm e}^{8 i x}}{4}-\frac {1}{4}\right ) f \polylog \left (4, -{\mathrm e}^{i x}\right )+\left (48 c_{4} x^{3}+48 x^{2} c_{3}+48 x c_{2}+48 c_{1}\right ) {\mathrm e}^{8 i x}+\left (-192 c_{4} x^{3}-192 x^{2} c_{3}-192 x c_{2}-192 c_{1}\right ) {\mathrm e}^{6 i x}+\left (288 c_{4} x^{3}+288 x^{2} c_{3}+288 x c_{2}+288 c_{1}\right ) {\mathrm e}^{4 i x}+\left (160 \arctanh \left ({\mathrm e}^{i x}\right ) f x +8 x^{3} \left (\csc ^{2}\left (x \right )+\frac {3}{2}\right ) \csc \left (x \right ) f \cot \left (x \right )-192 c_{4} x^{3}-192 x^{2} c_{3}-192 x c_{2}-192 c_{1}\right ) {\mathrm e}^{2 i x}-240 x \left (\arctanh \left ({\mathrm e}^{i x}\right )+\frac {x^{2} \left (\csc ^{2}\left (x \right )+\frac {3}{2}\right ) \csc \left (x \right ) \cot \left (x \right )}{20}\right ) f \,{\mathrm e}^{4 i x}+160 x \left (\arctanh \left ({\mathrm e}^{i x}\right )+\frac {x^{2} \left (\csc ^{2}\left (x \right )+\frac {3}{2}\right ) \csc \left (x \right ) \cot \left (x \right )}{20}\right ) f \,{\mathrm e}^{6 i x}-40 x \left (\arctanh \left ({\mathrm e}^{i x}\right )+\frac {x^{2} \left (\csc ^{2}\left (x \right )+\frac {3}{2}\right ) \csc \left (x \right ) \cot \left (x \right )}{20}\right ) f \,{\mathrm e}^{8 i x}-40 \arctanh \left ({\mathrm e}^{i x}\right ) f x +12 \left (\frac {11 x^{3}}{6}+i\right ) f \,{\mathrm e}^{3 i x}-12 \left (-\frac {11 x^{3}}{6}+i\right ) f \,{\mathrm e}^{5 i x}+4 \left (-\frac {3 x^{3}}{2}+i\right ) f \,{\mathrm e}^{7 i x}-4 \left (\frac {3 x^{3}}{2}+i\right ) f \,{\mathrm e}^{i x}-2 x^{3} \left (\csc ^{2}\left (x \right )+\frac {3}{2}\right ) \csc \left (x \right ) f \cot \left (x \right )+48 c_{4} x^{3}+48 x^{2} c_{3}+48 x c_{2}+48 c_{1}}{48 \left ({\mathrm e}^{2 i x}-1\right )^{4} \sin \left (x \right )}\]