4.3.41 $2y^{\prime }(x)+2{\csc }^2(x)=y(x)\csc (x)\sec (x)-y(x{)}^2{\sec }^2(x)$

ODE
$$2y^{\prime }(x)+2{\csc }^2(x)=y(x)\csc (x)\sec (x)-y(x{)}^2{\sec }^2(x)$$ ODE Classification

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica ✓
cpu = 0.476711 (sec), leaf count = 48

$$\left\{\{y(x)\to \frac{\cot (x)(c_1\sqrt{\cos (x)}+2\sqrt[4]{{\sin }^2(x)})}{c_1\sqrt{\cos (x)}+\sqrt[4]{{\sin }^2(x)}}\}\right\}$$

Maple ✓
cpu = 0.09 (sec), leaf count = 107

$$\{{\int }^{\frac{(\csc \left(x\right)\sec \left(x\right)+2\cot \left(x\right)+2\tan \left(x\right))y\left(x\right)}{2{(\csc \left(x\right))}^2}}{(\csc \left(x\right)\sec \left(x\right)+2\cot \left(x\right)+2\tan \left(x\right))}^2{(2({\mathit{\_}\mathit{a}}^2-\mathit{\_}\mathit{a}/2+1/2){(\sec \left(x\right))}^2{(\csc \left(x\right))}^2-4\sec \left(x\right)(\cot \left(x\right)+\tan \left(x\right))(\mathit{\_}\mathit{a}-1)\csc \left(x\right)-4{(\cot \left(x\right)+\tan \left(x\right))}^2(\mathit{\_}\mathit{a}-1))}^{-1}d\mathit{\_}\mathit{a}+\frac{\ln (\tan \left(x\right))}{2}+\ln (\sin \left(x\right))-\ln (\cos \left(x\right))+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[2*Csc[x]^2 + 2*y'[x] == Csc[x]*Sec[x]*y[x] - Sec[x]^2*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> (Cot[x]*(C[1]*Sqrt[Cos[x]] + 2*(Sin[x]^2)^(1/4)))/(C[1]*Sqrt[Cos[x]] +
 (Sin[x]^2)^(1/4))}}

Maple raw input

dsolve(2*diff(y(x),x)+2*csc(x)^2 = y(x)*csc(x)*sec(x)-y(x)^2*sec(x)^2, y(x),'implicit')

Maple raw output

Intat((csc(x)*sec(x)+2*cot(x)+2*tan(x))^2/(2*(_a^2-1/2*_a+1/2)*sec(x)^2*csc(x)^2
-4*sec(x)*(cot(x)+tan(x))*(_a-1)*csc(x)-4*(cot(x)+tan(x))^2*(_a-1)),_a = 1/2/csc
(x)^2*(csc(x)*sec(x)+2*cot(x)+2*tan(x))*y(x))+1/2*ln(tan(x))+ln(sin(x))-ln(cos(x
))+_C1 = 0