\[ y'(x)+y(x) \cos (x)-e^{-\sin (x)}=0 \] ✓ Mathematica : cpu = 0.0252519 (sec), leaf count = 16
\[\left \{\left \{y(x)\to \left (c_1+x\right ) e^{-\sin (x)}\right \}\right \}\]
✓ Maple : cpu = 0.007 (sec), leaf count = 13
\[ \left \{ y \left ( x \right ) = \left ( x+{\it \_C1} \right ) {{\rm e}^{-\sin \left ( x \right ) }} \right \} \]
\begin {equation} \frac {dy}{dx}+y\left ( x\right ) \cos \left ( x\right ) =e^{-\sin \left ( x\right ) }\tag {1} \end {equation}
Integrating factor \(\mu =e^{\int \cos dx}=e^{\sin x}\). Hence (1) becomes
\[ \frac {d}{dx}\left ( \mu y\left ( x\right ) \right ) =\mu e^{-\sin \left ( x\right ) }\]
Replacing \(\mu \) by \(e^{\sin x}\) and integrating both sides
\begin {align*} e^{\sin x}y\left ( x\right ) & =\int e^{\sin x}e^{-\sin \left ( x\right ) }dx+C\\ e^{\sin x}y\left ( x\right ) & =\int dx+C\\ e^{\sin x}y\left ( x\right ) & =x+C\\ y\left ( x\right ) & =xe^{-\sin x}+Ce^{-\sin \left ( x\right ) } \end {align*}