##### 4.1.21 $$y'(x)=4 x \csc (x) \sec ^2(x)-2 y(x) \cot (2 x)$$

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

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.201695 (sec), leaf count = 81

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

Maple
cpu = 1.021 (sec), leaf count = 98

$\left [y \left (x \right ) = \left (-32 \sqrt {-\frac {{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{4 i x}-1\right )^{2}}}\, \left (\frac {x \ln \left (1+i {\mathrm e}^{i x}\right )}{2}-\frac {x \ln \left (1-i {\mathrm e}^{i x}\right )}{2}-\frac {i \dilog \left (1+i {\mathrm e}^{i x}\right )}{2}+\frac {i \dilog \left (1-i {\mathrm e}^{i x}\right )}{2}\right ) \sin \left (2 x \right )+\textit {\_C1} \right ) \sqrt {\cot ^{2}\left (2 x \right )+1}\right ]$ 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))

Maple raw output

[y(x) = (-32*(-exp(4*I*x)/(exp(4*I*x)-1)^2)^(1/2)*(1/2*x*ln(1+I*exp(I*x))-1/2*x*
ln(1-I*exp(I*x))-1/2*I*dilog(1+I*exp(I*x))+1/2*I*dilog(1-I*exp(I*x)))*sin(2*x)+_
C1)*(cot(2*x)^2+1)^(1/2)]