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 = 0.0606009 (sec), leaf count = 81

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

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

\[ \left \{ y \relax (x ) =-16\, \left (\sin \left (2\,x \right ) \left (i{\it dilog} \left (1-i{{\rm e}^{ix}} \right ) -i{\it dilog} \left (1+i{{\rm e}^{ix}} \right ) -x\ln \left (1-i{{\rm e}^{ix}} \right ) +x\ln \left (1+i{{\rm e}^{ix}} \right ) \right ) \sqrt {-{\frac {{{\rm e}^{4\,ix}}}{ \left ({{\rm e}^{4\,ix}}-1 \right ) ^{2}}}}-{\it \_C1}/16 \right ) \sqrt { \left (\cot \left (2\,x \right ) \right ) ^{2}+1} \right \} \] 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),'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)