\[ y'(x)-y(x) (a+\sin (\log (x))+\cos (\log (x)))=0 \] ✓ Mathematica : cpu = 0.0152005 (sec), leaf count = 19
\[\left \{\left \{y(x)\to c_1 e^{a x+x \sin (\log (x))}\right \}\right \}\] ✓ Maple : cpu = 0.014 (sec), leaf count = 14
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{x \left ( \sin \left ( \ln \left ( x \right ) \right ) +a \right ) }} \right \} \]
\begin {equation} \frac {dy}{dx}-y\left ( x\right ) \left [ a+\sin \left ( \log \left ( x\right ) \right ) +\cos \left ( \log \left ( x\right ) \right ) \right ] =0\tag {1} \end {equation}
Integrating factor \(\mu =e^{-\int a-\sin \left ( \log \left ( x\right ) \right ) -\cos \left ( \log \left ( x\right ) \right ) dx}=e^{-ax}e^{-\int \sin \left ( \log \left ( x\right ) \right ) +\cos \left ( \log \left ( x\right ) \right ) dx}\). To integrate \(\int \sin \left ( \log \left ( x\right ) \right ) +\cos \left ( \log \left ( x\right ) \right ) dx\), let \(r=\log \left ( x\right ) \), \(\frac {dr}{dx}=\frac {1}{x}\), then \(dx=xdr\), But \(x=e^{r}\), hence the integral becomes
\begin {align} \int \sin \left ( \log \left ( x\right ) \right ) +\cos \left ( \log \left ( x\right ) \right ) dx & =\int \left [ \sin \left ( r\right ) +\cos \left ( r\right ) \right ] e^{r}dr\nonumber \\ & =\int e^{r}\sin \left ( r\right ) dr+\int e^{r}\cos \left ( r\right ) dr\tag {2} \end {align}
Integrating by parts \(\int e^{r}\cos \left ( r\right ) dr,\) \(\int udv=uv-\int vdu\), Let \(u=e^{r}\rightarrow du=e^{r}\) and \(dv=\cos \left ( r\right ) \rightarrow v=\sin \left ( r\right ) \), hence (2) becomes
\begin {align*} \int e^{r}\sin \left ( r\right ) dr+\int e^{r}\cos \left ( r\right ) dr & =\int e^{r}\sin \left ( r\right ) dr+e^{r}\sin \left ( r\right ) -\int \sin \left ( r\right ) e^{r}dr\\ & =e^{r}\sin \left ( r\right ) \end {align*}
Therefore, substituting back \(r=\log \left ( x\right ) \) gives
\begin {align*} \int \sin \left ( \log \left ( x\right ) \right ) +\cos \left ( \log \left ( x\right ) \right ) dx & =e^{\log \left ( x\right ) }\sin \left ( \log \left ( x\right ) \right ) \\ & =x\sin \left ( \log \left ( x\right ) \right ) \end {align*}
Hence the integration factor is
\begin {align*} \mu & =e^{-ax}e^{-\int \sin \left ( \log \left ( x\right ) \right ) +\cos \left ( \log \left ( x\right ) \right ) dx}\\ & =e^{-ax}e^{-x\sin \left ( \log \left ( x\right ) \right ) } \end {align*}
Therefore (1) becomes
\[ \frac {d}{dx}\left ( \mu y\left ( x\right ) \right ) =0 \]
Integrating
\begin {align*} y\left ( x\right ) e^{-ax}e^{-x\sin \left ( \log \left ( x\right ) \right ) } & =C\\ y\left ( x\right ) & =Ce^{ax}e^{x\sin \left ( \log \left ( x\right ) \right ) }\\ & =Ce^{ax+x\sin \left ( \log \left ( x\right ) \right ) }\\ & =Ce^{x\left ( a+\sin \left ( \log \left ( x\right ) \right ) \right ) } \end {align*}