4.43.49 $y^{‴}(x)+y^{″}(x)(2\cot (x)+\csc (x))-y^{\prime }(x)=\cot (x)$

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

[[_3rd_order, _missing_y]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.680945 (sec), leaf count = 56

$$\left\{\{y(x)\to -\frac{c_{2}x}{\sqrt{2}}-\frac{\cot \left(\frac{x}{2}\right)(c_2\log (2(\cos (x)+1))+2c_1)}{\sqrt{2}}+c_3+\cot \left(\frac{x}{2}\right){\sin }^{-1}(\cos (x))\}\right\}$$

Maple ✓
cpu = 28.87 (sec), leaf count = 276

$$\{y\left(x\right)=\frac{-\mathit{\_}\mathit{C}\mathit{1}\pi {\left(\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{\mathrm{e}}^{-ix}{({\mathrm{e}}^{ix}+1)}^2\right)\right)}^3+\mathit{\_}\mathit{C}\mathit{1}\pi (\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{\mathrm{e}}^{-ix}\right)+\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{({\mathrm{e}}^{ix}+1)}^2\right)){\left(\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{\mathrm{e}}^{-ix}{({\mathrm{e}}^{ix}+1)}^2\right)\right)}^2-\mathit{\_}\mathit{C}\mathit{1}\pi \mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{({\mathrm{e}}^{ix}+1)}^2\right)\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{\mathrm{e}}^{-ix}\right)\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{\mathrm{e}}^{-ix}{({\mathrm{e}}^{ix}+1)}^2\right)-\mathit{\_}\mathit{C}\mathit{1}\pi {\left(\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{({\mathrm{e}}^{ix}+1)}^2\right)\right)}^3+2\mathit{\_}\mathit{C}\mathit{1}\pi \mathit{c}\mathit{s}\mathit{g}\mathit{n}(i+i{\mathrm{e}}^{ix}){\left(\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{({\mathrm{e}}^{ix}+1)}^2\right)\right)}^2-\mathit{\_}\mathit{C}\mathit{1}\pi {\left(\mathit{c}\mathit{s}\mathit{g}\mathit{n}(i+i{\mathrm{e}}^{ix})\right)}^2\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{({\mathrm{e}}^{ix}+1)}^2\right)-2i\mathit{\_}\mathit{C}\mathit{1}({\mathrm{e}}^{ix}+1)\ln ({\mathrm{e}}^{ix}+1)+2i\mathit{\_}\mathit{C}\mathit{1}\ln \left({\mathrm{e}}^{ix}\right)+(-ix-2\mathit{\_}\mathit{C}\mathit{1}x+\mathit{\_}\mathit{C}\mathit{3}){\mathrm{e}}^{ix}+2i\ln \left(2\right)\mathit{\_}\mathit{C}\mathit{1}-ix-2i\mathit{\_}\mathit{C}\mathit{2}+2\mathit{\_}\mathit{C}\mathit{1}x-\mathit{\_}\mathit{C}\mathit{3}}{{\mathrm{e}}^{ix}-1}\}$$ Mathematica raw input

DSolve[-y'[x] + (2*Cot[x] + Csc[x])*y''[x] + y'''[x] == Cot[x],y[x],x]

Mathematica raw output

{{y[x] -> -((x*C[2])/Sqrt[2]) + C[3] + ArcSin[Cos[x]]*Cot[x/2] - (Cot[x/2]*(2*C[
1] + C[2]*Log[2*(1 + Cos[x])]))/Sqrt[2]}}

Maple raw input

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

Maple raw output

y(x) = (-_C1*Pi*csgn(I*exp(-I*x)*(exp(I*x)+1)^2)^3+_C1*Pi*(csgn(I*exp(-I*x))+csg
n(I*(exp(I*x)+1)^2))*csgn(I*exp(-I*x)*(exp(I*x)+1)^2)^2-_C1*Pi*csgn(I*(exp(I*x)+
1)^2)*csgn(I*exp(-I*x))*csgn(I*exp(-I*x)*(exp(I*x)+1)^2)-_C1*Pi*csgn(I*(exp(I*x)
+1)^2)^3+2*_C1*Pi*csgn(I+I*exp(I*x))*csgn(I*(exp(I*x)+1)^2)^2-_C1*Pi*csgn(I+I*ex
p(I*x))^2*csgn(I*(exp(I*x)+1)^2)-2*I*_C1*(exp(I*x)+1)*ln(exp(I*x)+1)+2*I*_C1*ln(
exp(I*x))+(-I*x-2*_C1*x+_C3)*exp(I*x)+2*I*ln(2)*_C1-I*x-2*I*_C2+2*_C1*x-_C3)/(ex
p(I*x)-1)