4.1.24 $y^{\prime }(x)=4x\csc (x)(y(x)-{\tan }^2(x)+1)$

ODE
$$y^{\prime }(x)=4x\csc (x)(y(x)-{\tan }^2(x)+1)$$ ODE Classification

[_linear]

Book solution method
Linear ODE

Mathematica ✓
cpu = 28.6501 (sec), leaf count = 155

$$\left\{\{y(x)\to \exp (4i{\text{Li}}_2(-e^{ix})-4i{\text{Li}}_2\left(e^{ix}\right)+4x(\log (1-e^{ix})-\log (1+e^{ix})))(c_1+{\int }_1^{x}4K[1]\cos (2K[1])\csc (K[1]){\sec }^2(K[1])\exp (-4i{\text{Li}}_2(-e^{iK[1]})+4i{\text{Li}}_2\left(e^{iK[1]}\right)+4K[1](\log (1+e^{iK[1]})-\log (1-e^{iK[1]})))dK[1])\}\right\}$$

Maple ✓
cpu = 1.476 (sec), leaf count = 125

$$\{y\left(x\right)={\mathrm{e}}^{-4i(\mathit{p}\mathit{o}\mathit{l}\mathit{y}\mathit{l}\mathit{o}\mathit{g}(2,{\mathrm{e}}^{ix})-\mathit{p}\mathit{o}\mathit{l}\mathit{y}\mathit{l}\mathit{o}\mathit{g}(2,-{\mathrm{e}}^{ix}))}{(1-{\mathrm{e}}^{ix})}^{4x}{(1+{\mathrm{e}}^{ix})}^{-4x}(4\int 4\frac{x{(1-{\mathrm{e}}^{ix})}^{-4x}{(1+{\mathrm{e}}^{ix})}^{4x}{\mathrm{e}}^{4i(\mathit{p}\mathit{o}\mathit{l}\mathit{y}\mathit{l}\mathit{o}\mathit{g}(2,{\mathrm{e}}^{ix})-\mathit{p}\mathit{o}\mathit{l}\mathit{y}\mathit{l}\mathit{o}\mathit{g}(2,-{\mathrm{e}}^{ix}))}(-\sin \left(3x\right)+\sin \left(x\right))}{-1+\cos \left(4x\right)}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})\}$$ Mathematica raw input

DSolve[y'[x] == 4*x*Csc[x]*(1 - Tan[x]^2 + y[x]),y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(diff(y(x),x) = 4*csc(x)*x*(1-tan(x)^2+y(x)), y(x),'implicit')

Maple raw output

y(x) = exp(-4*I*(polylog(2,exp(I*x))-polylog(2,-exp(I*x))))*(1-exp(I*x))^(4*x)*(
1+exp(I*x))^(-4*x)*(4*Int(4*x*(1-exp(I*x))^(-4*x)*(1+exp(I*x))^(4*x)*exp(4*I*(po
lylog(2,exp(I*x))-polylog(2,-exp(I*x))))*(-sin(3*x)+sin(x))/(-1+cos(4*x)),x)+_C1
)