4.3.11 $$y'(x)=a+b \cos (A x+B y(x))$$

ODE
$y'(x)=a+b \cos (A x+B y(x))$ ODE Classiﬁcation

[[_homogeneous, class C], _dAlembert]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.842269 (sec), leaf count = 102

$\left \{\left \{y(x)\to -\frac {A x+2 \tan ^{-1}\left (\frac {(B (a+b)+A) \tanh \left (\frac {(x-c_1) \left (B^2 \left (a^2-b^2\right )+2 a A B+A^2\right )}{2 \sqrt {-((B (a-b)+A) (B (a+b)+A))}}\right )}{\sqrt {-((B (a-b)+A) (B (a+b)+A))}}\right )}{B}\right \}\right \}$

Maple
cpu = 0.096 (sec), leaf count = 96

$\left [y \left (x \right ) = -\frac {A x +2 \arctan \left (\frac {\tan \left (\frac {\textit {\_C1} \sqrt {\left (a B +b B +A \right ) \left (a B -b B +A \right )}}{2}-\frac {x \sqrt {\left (a B +b B +A \right ) \left (a B -b B +A \right )}}{2}\right ) \sqrt {\left (a B +b B +A \right ) \left (a B -b B +A \right )}}{a B -b B +A}\right )}{B}\right ]$ Mathematica raw input

DSolve[y'[x] == a + b*Cos[A*x + B*y[x]],y[x],x]

Mathematica raw output

{{y[x] -> -((A*x + 2*ArcTan[((A + (a + b)*B)*Tanh[((A^2 + 2*a*A*B + (a^2 - b^2)*
B^2)*(x - C[1]))/(2*Sqrt[-((A + (a - b)*B)*(A + (a + b)*B))])])/Sqrt[-((A + (a -
 b)*B)*(A + (a + b)*B))]])/B)}}

Maple raw input

dsolve(diff(y(x),x) = a+b*cos(A*x+B*y(x)), y(x))

Maple raw output

[y(x) = -(A*x+2*arctan(tan(1/2*_C1*((B*a+B*b+A)*(B*a-B*b+A))^(1/2)-1/2*x*((B*a+B
*b+A)*(B*a-B*b+A))^(1/2))*((B*a+B*b+A)*(B*a-B*b+A))^(1/2)/(B*a-B*b+A)))/B]