ODE
\[ y'(x)=a+b \sin (y(x)) \] ODE Classification
[_quadrature]
Book solution method
Separable ODE, Independent variable missing
Mathematica ✓
cpu = 0.410948 (sec), leaf count = 52
\[\left \{\left \{y(x)\to 2 \tan ^{-1}\left (\frac {-b+\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \sqrt {a^2-b^2} (x+c_1)\right )}{a}\right )\right \}\right \}\]
Maple ✓
cpu = 0.048 (sec), leaf count = 56
\[\left [y \left (x \right ) = -2 \arctan \left (\frac {-\tan \left (\frac {\textit {\_C1} \sqrt {a^{2}-b^{2}}}{2}+\frac {x \sqrt {a^{2}-b^{2}}}{2}\right ) \sqrt {a^{2}-b^{2}}+b}{a}\right )\right ]\] Mathematica raw input
DSolve[y'[x] == a + b*Sin[y[x]],y[x],x]
Mathematica raw output
{{y[x] -> 2*ArcTan[(-b + Sqrt[a^2 - b^2]*Tan[(Sqrt[a^2 - b^2]*(x + C[1]))/2])/a]}}
Maple raw input
dsolve(diff(y(x),x) = a+b*sin(y(x)), y(x))
Maple raw output
[y(x) = -2*arctan((-tan(1/2*_C1*(a^2-b^2)^(1/2)+1/2*x*(a^2-b^2)^(1/2))*(a^2-b^2)^(1/2)+b)/a)]