##### 4.13.27 $$y'(x) \left (\cot (x)-2 y(x)^2\right )=y(x)^3 \csc (x) \sec (x)$$

ODE
$y'(x) \left (\cot (x)-2 y(x)^2\right )=y(x)^3 \csc (x) \sec (x)$ ODE Classiﬁcation

[y=_G(x,y')]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.744601 (sec), leaf count = 69

$\left \{\left \{y(x)\to -\frac {i \sqrt {\cot (x)} \sqrt {W\left (-2 e^{-8 c_1} \tan (x)\right )}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {i \sqrt {\cot (x)} \sqrt {W\left (-2 e^{-8 c_1} \tan (x)\right )}}{\sqrt {2}}\right \}\right \}$

Maple
cpu = 100.217 (sec), leaf count = 0 , could not solve

dsolve((cot(x)-2*y(x)^2)*diff(y(x),x) = y(x)^3*csc(x)*sec(x), y(x))

Mathematica raw input

DSolve[(Cot[x] - 2*y[x]^2)*y'[x] == Csc[x]*Sec[x]*y[x]^3,y[x],x]

Mathematica raw output

{{y[x] -> ((-I)*Sqrt[Cot[x]]*Sqrt[ProductLog[(-2*Tan[x])/E^(8*C[1])]])/Sqrt[2]},
 {y[x] -> (I*Sqrt[Cot[x]]*Sqrt[ProductLog[(-2*Tan[x])/E^(8*C[1])]])/Sqrt[2]}}

Maple raw input

dsolve((cot(x)-2*y(x)^2)*diff(y(x),x) = y(x)^3*csc(x)*sec(x), y(x))

Maple raw output

dsolve((cot(x)-2*y(x)^2)*diff(y(x),x) = y(x)^3*csc(x)*sec(x), y(x))