I was reading the list of integrals from Wikipedia, and I've created a variation of the integral $$\int \log_a(x)dx.$$ I show my example as a definite integral. I would like to know if is it possible to find an approximation of the integal of such logarithm in my example
$$\int_0^1\frac{\log x}{\log(\gamma+\psi(x+1))}dx,\tag{1}$$ where $\gamma$ is the Euler-Mascheroni constant and $\psi(s)$ the digamma function.
Question. How can we justify an approximation of $(1)$? Preferably using analysis and some asymptotic property of functions in the integrand. If this integral was in the literature and you need answer it as a reference request feel free to do it (then I am going to search and try understand such literaure). Many thanks.
See the integration that I've encoded and my integrand with Wolfram Alpha online calculator
int log(x)/log(EulerGamma+Digamma(x+1))dx, from x=0 to 1
plot log(x)/log(EulerGamma+Digamma(x+1)), from x=0 to 1
This is not an answer to the question.
Since I wrote it in a comment, let us consider something as simple as $$I=\int_0^1 \frac{\log (x+1)}{\log (x+2)}\,dx$$ which, more than likely, does not show a closed form.
The numerical integration leads to $\approx \color{red} {0.402665}$.
To approximate the result, I used $[n,2]$ Padé approximants built around $x=0$ ; these are "easy" to obtain. For example, for $n=2$, we have $$\frac{\log (x+1)}{\log (x+2)}=\frac{\frac{x}{\log (2)}+\frac{x^2}{2 \log (2)} } {1+x \left(1+\frac{1}{2 \log (2)}\right)+x^2 \left(\frac{1}{6}+\frac{3}{8 \log (2)}\right) }$$ Plot the two functions to see how close they are.
Integration does not make any problem and the result evaluates $\approx \color{red} {0.402}705$ which is not too bad.
Doing the same with $n=3$ would lead to $\approx \color{red} {0.4026}52$.
Now, I give up !