How to integrate $\frac{1}{x\ln(1+x)}$?

Question

I tried to evaluate the indefinite integral

$$\int{\frac{dx}{x\ln(1+x)}}.$$

Even Wolfram Alpha can't give any result in terms of standard mathematical functions. Here are some related integrals:

$$\int{\frac{\ln(x)}{\ln(1+x)}}\,dx$$

$$\int{\frac{Li(x)}{1+x}}\,dx,$$

Unfortunately, I'm not able to solve any of them. Is there any special function related with this kind of integrals?

Answer 1

$$I=\int\frac{dx}{x\ln(1+x)}$$ let $u=\ln(1+x)$ so $du=\frac{1}{x+1}dx\Rightarrow dx=e^udu$ and $x=e^u-1$ so: $$I=\int\frac{e^udu}{(e^u-1)u}=\int\frac{du}{u}+\int\frac{du}{(e^u-1)u}$$ the first integral is obviously quite easy and you can note that: $$\frac{1}{(e^u-1)}=\frac{e^{-u}}{1-e^{-u}}$$ which might look familiar to you as the sum of a geometric series, try this out and see if it comes to anything :)

Answer 2

Probably dangerous

For an approximation, expand $\frac{1}{\log (a+x)}$ as a Taylor series around $a=0$ and later make $a=1$.

This will give $$\frac{1}{x\log (1+x)}=\sum_{n=1}^p (-1)^{n+1}\frac {P_n(t)}{b_n\,x^n} \qquad \text {where} \qquad t=\frac 1{\log(x)}$$

The first $b_n$ are $$\{1,1,3,12,60,360,210,1680,15120,151200,\cdots\}$$ and the very first polynomials are $$\left( \begin{array}{cc} n & P_n(t) \\ 1 & t \\ 2 & t^2 \\ 3 & t^2 (2 t+1) \\ 4 & t^2 \left(3 t^2+3 t+1\right) \\ 5 & t^2 \left(12 t^3+18 t^2+11 t+3\right) \\ 6 & t^2 \left(60 t^4+120 t^3+105 t^2+50 t+12\right) \\ 7 & t^2 \left(360 t^5+900 t^4+1020 t^3+675 t^2+274 t+60\right) \\ 8 & t^2 \left(210 t^6+630 t^5+875 t^4+735 t^3+406 t^2+147 t+30\right) \end{array} \right)$$ Now $$\int \frac {dx} {x^p \log^q(x)}=\int e^{(1-p) y} y^{-q}\,dy=-(p-1)^{q-1} \Gamma (1-q,(p-1) y) $$

For a quick check, using $p=10$, I computed $$I_n=\int_e^{e^n} \frac{dx}{x \log (x+1)}$$ $$\left( \begin{array}{ccc} n & \text{approximation} & \text{"exact"} \\ 2 & 0.59692 & 0.59703 \\ 3 & 0.98844 & 0.98854 \\ 4 & 1.27345 & 1.27355 \\ 5 & 1.49600 & 1.49610 \\ 6 & 1.67817 & 1.67827 \\ 7 & 1.83228 & 1.83239 \\ 8 & 1.96581 & 1.96591 \\ 9 & 2.08359 & 2.08369 \\ 10 & 2.18895 & 2.18905\\ 20 & 2.88209 & 2.88219 \\ 30 & 3.28756 & 3.28766 \\ 40 & 3.57524 & 3.57534 \\ 50 & 3.79485 & 3.79494 \end{array} \right)$$