4.1.21 $y^{\prime }(x)=4x\csc (x){\sec }^2(x)-2y(x)\cot (2x)$

ODE
$$y^{\prime }(x)=4x\csc (x){\sec }^2(x)-2y(x)\cot (2x)$$ ODE Classification

[_linear]

Book solution method
Linear ODE

Mathematica ✓
cpu = 0.0606009 (sec), leaf count = 81

$$\left\{\{y(x)\to \csc (x)\sec (x)(c_1+4i{\text{Li}}_2(-i{e}^{ix})-4i{\text{Li}}_2\left(i{e}^{ix}\right)+4x\log (1-i{e}^{ix})-4x\log (1+i{e}^{ix}))\}\right\}$$

Maple ✓
cpu = 1.581 (sec), leaf count = 99

$$\{y\left(x\right)=-16(\sin \left(2x\right)(i\mathit{d}\mathit{i}\mathit{l}\mathit{o}\mathit{g}(1-i{\mathrm{e}}^{ix})-i\mathit{d}\mathit{i}\mathit{l}\mathit{o}\mathit{g}(1+i{\mathrm{e}}^{ix})-x\ln (1-i{\mathrm{e}}^{ix})+x\ln (1+i{\mathrm{e}}^{ix}))\sqrt{-\frac{{\mathrm{e}}^{4ix}}{{({\mathrm{e}}^{4ix}-1)}^2}}-\mathit{\_}\mathit{C}\mathit{1}/16)\sqrt{{(\cot \left(2x\right))}^2+1}\}$$ Mathematica raw input

DSolve[y'[x] == 4*x*Csc[x]*Sec[x]^2 - 2*Cot[2*x]*y[x],y[x],x]

Mathematica raw output

{{y[x] -> Csc[x]*(C[1] + 4*x*Log[1 - I*E^(I*x)] - 4*x*Log[1 + I*E^(I*x)] + (4*I)
*PolyLog[2, (-I)*E^(I*x)] - (4*I)*PolyLog[2, I*E^(I*x)])*Sec[x]}}

Maple raw input

dsolve(diff(y(x),x) = 4*csc(x)*x*sec(x)^2-2*y(x)*cot(2*x), y(x),'implicit')

Maple raw output

y(x) = -16*(sin(2*x)*(I*dilog(1-I*exp(I*x))-I*dilog(1+I*exp(I*x))-x*ln(1-I*exp(I
*x))+x*ln(1+I*exp(I*x)))*(-exp(4*I*x)/(exp(4*I*x)-1)^2)^(1/2)-1/16*_C1)*(cot(2*x
)^2+1)^(1/2)