##### 4.10.12 $$y'(x) (y(x)-\cot (x) \csc (x))+y(x) \csc (x) (y(x) \cos (x)+1)=0$$

ODE
$y'(x) (y(x)-\cot (x) \csc (x))+y(x) \csc (x) (y(x) \cos (x)+1)=0$ ODE Classiﬁcation

[[_Abel, 2nd type, class A]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.576237 (sec), leaf count = 85

$\left \{\left \{y(x)\to \cot (x) \csc (x)-\frac {i \csc ^2(x) \sqrt {(-1+c_1) \cos (2 x)-1-c_1}}{\sqrt {2}}\right \},\left \{y(x)\to \cot (x) \csc (x)+\frac {i \csc ^2(x) \sqrt {(-1+c_1) \cos (2 x)-1-c_1}}{\sqrt {2}}\right \}\right \}$

Maple
cpu = 0.151 (sec), leaf count = 70

$\left [y \left (x \right ) = -\frac {-\sin \left (x \right ) \sqrt {\frac {\textit {\_C1} \left (\sin ^{2}\left (x \right )\right )+1}{\sin \left (x \right )^{2}}}+\cos \left (x \right )}{\cos ^{2}\left (x \right )-1}, y \left (x \right ) = -\frac {\sin \left (x \right ) \sqrt {\frac {\textit {\_C1} \left (\sin ^{2}\left (x \right )\right )+1}{\sin \left (x \right )^{2}}}+\cos \left (x \right )}{\cos ^{2}\left (x \right )-1}\right ]$ Mathematica raw input

DSolve[Csc[x]*y[x]*(1 + Cos[x]*y[x]) + (-(Cot[x]*Csc[x]) + y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> Cot[x]*Csc[x] - (I*Sqrt[-1 - C[1] + (-1 + C[1])*Cos[2*x]]*Csc[x]^2)/Sq
rt[2]}, {y[x] -> Cot[x]*Csc[x] + (I*Sqrt[-1 - C[1] + (-1 + C[1])*Cos[2*x]]*Csc[x
]^2)/Sqrt[2]}}

Maple raw input

dsolve((y(x)-cot(x)*csc(x))*diff(y(x),x)+csc(x)*(1+y(x)*cos(x))*y(x) = 0, y(x))

Maple raw output

[y(x) = -(-sin(x)*((_C1*sin(x)^2+1)/sin(x)^2)^(1/2)+cos(x))/(cos(x)^2-1), y(x) =
 -(sin(x)*((_C1*sin(x)^2+1)/sin(x)^2)^(1/2)+cos(x))/(cos(x)^2-1)]