\[ a \sin (\alpha y(x)+\beta x)+b+y'(x)=0 \] ✓ Mathematica : cpu = 0.68662 (sec), leaf count = 1317
\[\left \{\left \{y(x)\to \frac {2 \tan ^{-1}\left (\frac {\frac {a^2 \sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))} \tan \left (\frac {1}{2} \left (\frac {a^2 x \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {b^2 x \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {a^2 c_1 \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {b^2 c_1 \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {2 b \beta x \alpha }{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {2 b \beta c_1 \alpha }{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {\beta ^2 x}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {\beta ^2 c_1}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}\right )\right ) \alpha ^2}{(a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta )}-\frac {b^2 \sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))} \tan \left (\frac {1}{2} \left (\frac {a^2 x \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {b^2 x \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {a^2 c_1 \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {b^2 c_1 \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {2 b \beta x \alpha }{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {2 b \beta c_1 \alpha }{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {\beta ^2 x}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {\beta ^2 c_1}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}\right )\right ) \alpha ^2}{(a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta )}-a \alpha +\frac {2 b \beta \sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))} \tan \left (\frac {1}{2} \left (\frac {a^2 x \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {b^2 x \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {a^2 c_1 \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {b^2 c_1 \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {2 b \beta x \alpha }{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {2 b \beta c_1 \alpha }{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {\beta ^2 x}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {\beta ^2 c_1}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}\right )\right ) \alpha }{(a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta )}-\frac {\beta ^2 \sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))} \tan \left (\frac {1}{2} \left (\frac {a^2 x \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {b^2 x \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {a^2 c_1 \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {b^2 c_1 \alpha ^2}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {2 b \beta x \alpha }{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {2 b \beta c_1 \alpha }{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}-\frac {\beta ^2 x}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}+\frac {\beta ^2 c_1}{\sqrt {-((a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta ))}}\right )\right )}{(a \alpha +b \alpha -\beta ) (a \alpha -b \alpha +\beta )}}{\alpha b-\beta }\right )-\beta x}{\alpha }\right \}\right \}\] ✓ Maple : cpu = 0.109 (sec), leaf count = 89
\[y \relax (x ) = \frac {-\beta x +2 \arctan \left (\frac {-\tan \left (\frac {\sqrt {\left (-a^{2}+b^{2}\right ) \alpha ^{2}-2 \alpha b \beta +\beta ^{2}}\, \left (x -c_{1}\right )}{2}\right ) \sqrt {\left (-a^{2}+b^{2}\right ) \alpha ^{2}-2 \alpha b \beta +\beta ^{2}}-a \alpha }{b \alpha -\beta }\right )}{\alpha }\]
Hand solution
\[ y^{\prime }=-a\sin \left (\alpha y+\beta x\right ) -b \]
This is separable after transformation of \(u=\alpha y+\beta x\), hence \(u^{\prime }=\alpha y^{\prime }+\beta \) or \(y^{\prime }=\frac {1}{\alpha }\left ( u^{\prime }-\beta \right ) \). Therefore the above becomes\begin {align} \frac {1}{\alpha }\left (u^{\prime }-\beta \right ) & =-a\sin \relax (u) -b\nonumber \\ u^{\prime } & =-\alpha \left (a\sin \relax (u) +b\right ) +\beta \nonumber \\ \frac {du}{\beta -\alpha \left (a\sin \relax (u) +b\right ) } & =dx\tag {1} \end {align}
Using half angle tan transformation where \(\tan \left (\frac {u}{2}\right ) =t,\sin \relax (u) =\frac {2t}{t^{2}+1},du=\frac {2}{1+t^{2}}dt\) then\begin {align*} \int \frac {du}{\beta -\alpha \left (a\sin \relax (u) +b\right ) } & =\int \frac {2}{1+t^{2}}\frac {dt}{\beta -\alpha \left (a\frac {2t}{t^{2}+1}+b\right ) }\\ & =2\int \frac {dt}{\beta \left (t^{2}+1\right ) -\alpha \left (a2t+b\left ( t^{2}+1\right ) \right ) }\\ & =2\int \frac {dt}{t\beta ^{2}+\beta -\left (\alpha a2t+t^{2}\alpha b+\alpha b\right ) }\\ & =2\int \frac {dt}{\left (\beta -\alpha b\right ) \left (\frac {t\beta ^{2}-\alpha a2t-t^{2}\alpha b}{\left (\beta -\alpha b\right ) }+1\right ) }\\ & =\frac {2}{\left (\beta -\alpha b\right ) }\int \frac {dt}{\frac {t\beta ^{2}-\alpha a2t-t^{2}\alpha b}{\left (\beta -\alpha b\right ) }+1}\\ & =\frac {2}{\left (\beta -\alpha b\right ) }\frac {-\left (\alpha b-\beta \right ) }{\sqrt {\alpha ^{2}a^{2}-\left (\alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\tanh ^{-1}\left (\frac {\left (t+\frac {a\alpha }{b\alpha -\beta }\right ) \left (b\alpha -\beta \right ) }{\sqrt {\alpha ^{2}a^{2}-\left (\alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\right ) \\ & =\frac {2}{\sqrt {\alpha ^{2}a^{2}-\left (\alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\tanh ^{-1}\left (\frac {t\left (b\alpha -\beta \right ) +a\alpha }{\sqrt {\alpha ^{2}a^{2}-\left (\alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\right ) \end {align*}
But \(t=\tan \left (\frac {u}{2}\right ) \) therefore\[ \int \frac {du}{\beta -\alpha \left (a\sin \relax (u) +b\right ) }=\frac {2}{\sqrt {\alpha ^{2}a^{2}-\left (\alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\tanh ^{-1}\left (\frac {\tan \left (\frac {u}{2}\right ) \left (b\alpha -\beta \right ) +a\alpha }{\sqrt {\alpha ^{2}a^{2}-\left ( \alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\right ) \] But \(u=\alpha y+\beta x\), and the above becomes\[ \int \frac {du}{\beta -\alpha \left (a\sin \relax (u) +b\right ) }=\frac {2\tanh ^{-1}\left (\frac {\tan \left (\frac {\alpha y+\beta x}{2}\right ) \left (b\alpha -\beta \right ) +a\alpha }{\sqrt {\alpha ^{2}a^{2}-\left ( \alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\right ) }{\sqrt {\alpha ^{2}a^{2}-\left (\alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\] Back to (1), therefore after integrating both sides\[ \frac {2\tanh ^{-1}\left (\frac {\tan \left (\frac {\alpha y+\beta x}{2}\right ) \left (b\alpha -\beta \right ) +a\alpha }{\sqrt {\alpha ^{2}a^{2}-\left ( \alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}\right ) }{\sqrt {\alpha ^{2}a^{2}-\left (\alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }}=x+C \]
Let \[ A=\sqrt {\alpha ^{2}a^{2}-\left (\alpha ^{2}b^{2}+\beta ^{2}-2\alpha b\beta \right ) }\] Then
\begin {align*} \tanh ^{-1}\left (\frac {\tan \left (\frac {\alpha y+\beta x}{2}\right ) \left ( b\alpha -\beta \right ) +a\alpha }{A}\right ) & =\frac {1}{2}A\left ( x+C\right ) \\ \frac {\tan \left (\frac {\alpha y+\beta x}{2}\right ) \left (b\alpha -\beta \right ) +a\alpha }{A} & =\tanh \left (\frac {1}{2}A\left (x+C\right ) \right ) \\ \tan \left (\frac {\alpha y+\beta x}{2}\right ) \left (b\alpha -\beta \right ) +a\alpha & =A\tanh \left (\frac {1}{2}A\left (x+C\right ) \right ) \\ \tan \left (\frac {\alpha y+\beta x}{2}\right ) & =\frac {A}{\left ( b\alpha -\beta \right ) }\tanh \left (\frac {1}{2}A\left (x+C\right ) \right ) -\frac {a\alpha }{\left (b\alpha -\beta \right ) }\\ \frac {\alpha y+\beta x}{2} & =\arctan \left (\frac {A}{\left (b\alpha -\beta \right ) }\tanh \left (\frac {1}{2}A\left (x+C\right ) \right ) -\frac {a\alpha }{\left (b\alpha -\beta \right ) }\right ) \\ y & =\frac {2}{\alpha }\arctan \left (\frac {A}{\left (b\alpha -\beta \right ) }\tanh \left (\frac {1}{2}A\left (x+C\right ) \right ) -\frac {a\alpha }{\left ( b\alpha -\beta \right ) }\right ) -\frac {\beta x}{\alpha } \end {align*}
Verification