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))]