Primitive means integration

$\Rightarrow \int \cos \left(\log x\right)dx=f\left(x\right)\left\{\cos \left(g\left(x\right)\right)+\sin \left(h\left(x\right)\right)\right\}$

$\Rightarrow I=\int \cos \left(\log x\right)dx$

By parts formula

$\int u·v dx=u\int v dx-\int \left(\frac{d}{dx}\left(u\right)\int vdx\right)dx$

$\Rightarrow I=\int 1.\cos \left(\log x\right)dx$

$\Rightarrow I=\cos \left(\log x\right)·x-\int \left(\frac{-\sin \left(\log x\right)}{x}\right)x dx$

$\Rightarrow I=x\cos \left(\log x\right)+\int \sin \left(\log x\right)dx$

$\Rightarrow I=x\cos \left(\log x\right)+\left[\sin \left(\log x\right)x-\int \frac{\cos \left(\log x\right)}{x}x dx\right]$

$\Rightarrow I=x\cos \left(\log x\right)+x\sin \left(\log x\right)-\int \cos \left(\log x\right)dx$

$\Rightarrow 2I=x\left[\cos \left(\log x\right)+\sin \left(\log x\right)\right]+c \left[\because I=\int \cos \left(\log x\right)dx\right]$

$\Rightarrow I=\frac{x}{2}\left[\cos \left(\log x\right)+\sin \left(\log x\right)\right]+C$

$\Rightarrow I=f\left(x\right)\left[\cos \left(\log x\right)+\sin \left(\log x\right)\right]+C$

$\Rightarrow f\left(x\right)=\frac{x}{2},g\left(x\right)=\log x, h\left(x\right)=\log x$

$\Rightarrow f^{\text{'}}\left(x\right)=\frac{1}{2}$