Quotient Rule for Derivatives

Using the quotient rule in reverse, find the value of $$\int_{1}^{e} \frac{\ln(x)}{(1+\ln(x))^2} \, dx.$$

Please answer in the form of $\frac{a}{b}-c$, where $a, b, c$ are real numbers and $\gcd(a, b)=1$.

Source: ASMT 2016 Calculus #9