$\int \frac{f(x) g'(x)+f'(x) g(x)}{f(x) g(x)}\{\log f(x)+\log g(x)\} d x$ is equal to

$\frac{1}{2}[\log \{f(x) g(x)\}]^2+C$

Let

$I =\int \frac{f(x) g'(x)+f'(x) g(x)}{f(x) g(x)}\{\log f(x)+\log g(x)\} d x$

$\Rightarrow I =\int \log \{f(x) g(x)\} \times \frac{1}{f(x) g(x)} d\{f(x) g(x)\}$

$\Rightarrow I =\int \log \{f(x) g(x)\} d[\log \{f(x) g(x)\}]$

$\Rightarrow I=\frac{1}{2}[\log \{f(x) g(x)\}]^2+C$