##### 4.5.13 $$x y'(x)+\tan (y(x)+x)+x=0$$

ODE
$x y'(x)+\tan (y(x)+x)+x=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.373819 (sec), leaf count = 16

$\left \{\left \{y(x)\to -x+\sin ^{-1}\left (\frac {c_1}{x}\right )\right \}\right \}$

Maple
cpu = 0.15 (sec), leaf count = 166

$\left [y \left (x \right ) = \arctan \left (\frac {\sin \left (x \right ) \cos \left (x \right ) \textit {\_C1} \,x^{2}+\sqrt {\left (\sin ^{4}\left (x \right )\right ) \textit {\_C1}^{2} x^{4}+\left (\sin ^{2}\left (x \right )\right ) \left (\cos ^{2}\left (x \right )\right ) \textit {\_C1}^{2} x^{4}-\left (\sin ^{2}\left (x \right )\right ) \textit {\_C1}^{2} x^{4}+x^{2} \textit {\_C1} -1}}{\left (\sin ^{2}\left (x \right )\right ) \textit {\_C1} \,x^{2}-x^{2} \textit {\_C1} +1}\right ), y \left (x \right ) = -\arctan \left (\frac {-\sin \left (x \right ) \cos \left (x \right ) \textit {\_C1} \,x^{2}+\sqrt {\left (\sin ^{4}\left (x \right )\right ) \textit {\_C1}^{2} x^{4}+\left (\sin ^{2}\left (x \right )\right ) \left (\cos ^{2}\left (x \right )\right ) \textit {\_C1}^{2} x^{4}-\left (\sin ^{2}\left (x \right )\right ) \textit {\_C1}^{2} x^{4}+x^{2} \textit {\_C1} -1}}{\left (\sin ^{2}\left (x \right )\right ) \textit {\_C1} \,x^{2}-x^{2} \textit {\_C1} +1}\right )\right ]$ Mathematica raw input

DSolve[x + Tan[x + y[x]] + x*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -x + ArcSin[C[1]/x]}}

Maple raw input

dsolve(x*diff(y(x),x)+x+tan(x+y(x)) = 0, y(x))

Maple raw output

[y(x) = arctan((sin(x)*cos(x)*_C1*x^2+(sin(x)^4*_C1^2*x^4+sin(x)^2*cos(x)^2*_C1^
2*x^4-sin(x)^2*_C1^2*x^4+x^2*_C1-1)^(1/2))/(sin(x)^2*_C1*x^2-x^2*_C1+1)), y(x) =
 -arctan((-sin(x)*cos(x)*_C1*x^2+(sin(x)^4*_C1^2*x^4+sin(x)^2*cos(x)^2*_C1^2*x^4
-sin(x)^2*_C1^2*x^4+x^2*_C1-1)^(1/2))/(sin(x)^2*_C1*x^2-x^2*_C1+1))]