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 = 0.0502895 (sec), leaf count = 85

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

Maple
cpu = 0.111 (sec), leaf count = 74

\[ \left \{ -{\frac { \left (\sin \relax (x ) \right ) ^{2}}{ \left (\cos \relax (x ) \right ) ^{2}y \relax (x ) +\cos \relax (x ) -y \relax (x ) }}-{\sin \relax (x ) {\frac {1}{\sqrt {{\it \_C1}+ \left (\sin \relax (x ) \right ) ^{-2}}}}}=0,-{\frac { \left (\sin \relax (x ) \right ) ^{2}}{ \left (\cos \relax (x ) \right ) ^{2}y \relax (x ) +\cos \relax (x ) -y \relax (x ) }}+{\sin \relax (x ) {\frac {1}{\sqrt {{\it \_C1}+ \left (\sin \relax (x ) \right ) ^{-2}}}}}=0 \right \} \] 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),'implicit')

Maple raw output

-sin(x)^2/(cos(x)^2*y(x)+cos(x)-y(x))+sin(x)/(_C1+1/sin(x)^2)^(1/2) = 0, -sin(x)
^2/(cos(x)^2*y(x)+cos(x)-y(x))-sin(x)/(_C1+1/sin(x)^2)^(1/2) = 0