ODE
\[ y'(x)=a+b \sin (A x+B y(x)) \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
Change of Variable, new dependent variable
Mathematica ✓
cpu = 0.87412 (sec), leaf count = 94
\[\left \{\left \{y(x)\to \frac {-A x+2 \tan ^{-1}\left (\frac {-b B+\sqrt {B^2 \left (a^2-b^2\right )+2 a A B+A^2} \tan \left (\frac {1}{2} (x-c_1) \sqrt {B^2 \left (a^2-b^2\right )+2 a A B+A^2}\right )}{a B+A}\right )}{B}\right \}\right \}\]
Maple ✓
cpu = 0.1 (sec), leaf count = 115
\[\left [y \left (x \right ) = -\frac {A x +2 \arctan \left (\frac {b B +\tan \left (\frac {\textit {\_C1} \sqrt {B^{2} a^{2}-b^{2} B^{2}+2 A B a +A^{2}}}{2}-\frac {x \sqrt {B^{2} a^{2}-b^{2} B^{2}+2 A B a +A^{2}}}{2}\right ) \sqrt {B^{2} a^{2}-b^{2} B^{2}+2 A B a +A^{2}}}{a B +A}\right )}{B}\right ]\] Mathematica raw input
DSolve[y'[x] == a + b*Sin[A*x + B*y[x]],y[x],x]
Mathematica raw output
{{y[x] -> (-(A*x) + 2*ArcTan[(-(b*B) + Sqrt[A^2 + 2*a*A*B + (a^2 - b^2)*B^2]*Tan[(Sqrt[A^2 + 2*a*A*B + (a^2 - b^2)*B^2]*(x - C[1]))/2])/(A + a*B)])/B}}
Maple raw input
dsolve(diff(y(x),x) = a+b*sin(A*x+B*y(x)), y(x))
Maple raw output
[y(x) = -(A*x+2*arctan((b*B+tan(1/2*_C1*(B^2*a^2-B^2*b^2+2*A*B*a+A^2)^(1/2)-1/2*x*(B^2*a^2-B^2*b^2+2*A*B*a+A^2)^(1/2))*(B^2*a^2-B^2*b^2+2*A*B*a+A^2)^(1/2))/(B*a+A)))/B]