I can't think of a reason why this may be false.
If I have a linear function $f(x)$ which is strictly increasing (assume $f(x) \neq 0$ and NOT OTHER DISCONTINUTIES ) on a domain $D$, then
$\int_D \dfrac{1}{f(x)} dx = \ln(f(x)) \bigg|_{D}$
No absolute value signs because f(x) is monotonically increasing.
I know it's a rather big claim and I tested it out on $\dfrac{1}{1+x}$ and other similarly simple cases and it see to work.
EDIT: I just noticed $\dfrac{1}{1+x}$ also happens to be monotonically increasing on $(0,\infty)$
EDIT2: Let's tweak the conditions on $D$. Assume x > 0