##### 4.3.12 $$y'(x)=-\left (1-f'(x)\right ) \cos (y(x))+f'(x)-f(x) \sin (y(x))+1$$

ODE
$y'(x)=-\left (1-f'(x)\right ) \cos (y(x))+f'(x)-f(x) \sin (y(x))+1$ ODE Classiﬁcation

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Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.286206 (sec), leaf count = 59

$\left \{\left \{y(x)\to 2 \tan ^{-1}\left (f(x)+\frac {\exp \left (-\int _1^x-f(K[1])dK[1]\right )}{\int _1^x-\exp \left (-\int _1^{K[2]}-f(K[1])dK[1]\right )dK[2]+c_1}\right )\right \}\right \}$

Maple
cpu = 1.516 (sec), leaf count = 41

$\left [y \left (x \right ) = 2 \arctan \left (\frac {-{\mathrm e}^{\int f \left (x \right )d x}+\left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right ) f \left (x \right )+\textit {\_C1} f \left (x \right )}{\textit {\_C1} +\int {\mathrm e}^{\int f \left (x \right )d x}d x}\right )\right ]$ Mathematica raw input

DSolve[y'[x] == 1 - f[x]*Sin[y[x]] - Cos[y[x]]*(1 - f'[x]) + f'[x],y[x],x]

Mathematica raw output

{{y[x] -> 2*ArcTan[f[x] + 1/(E^Inactive[Integrate][-f[K[1]], {K[1], 1, x}]*(C[1]
 + Inactive[Integrate][-E^(-Inactive[Integrate][-f[K[1]], {K[1], 1, K[2]}]), {K[
2], 1, x}]))]}}

Maple raw input

dsolve(diff(y(x),x) = 1+diff(f(x),x)-f(x)*sin(y(x))-(1-diff(f(x),x))*cos(y(x)), y(x))

Maple raw output

[y(x) = 2*arctan((-exp(Int(f(x),x))+Int(exp(Int(f(x),x)),x)*f(x)+_C1*f(x))/(_C1+
Int(exp(Int(f(x),x)),x)))]