Answer
$\ln|\ln(x+1)|+C$
Work Step by Step
We will solve the given integral by using u-substitution method.
Let us consider that $u=\ln (x+1) \implies du=\dfrac{ \ dx}{x+1}$
$\displaystyle \int \dfrac{1}{(x+1) \ln (x+1)} \ dx=\int \displaystyle \dfrac{1}{\ln (x+1) (x+1)} \ dx$
or, $=\int \dfrac{1}{u} \ du$
or, $=\ln |u|+C$
Now, we will use back substitution $u=\ln (x+1)$
Therefore, we have:
$\displaystyle \int \dfrac{1}{(x+1) \ln (x+1)} \ dx=\ln|\ln(x+1)|+C$
