indefinite integral in terms of special functions

Question

I have encountered the following integral $$ I=\int \frac{1}{\log(x)+x} dx $$

I know that the integral is not elementary. Is it possible to express the antiderivative in terms of known special functions?

Answer 1

I do not know any special function for this integral. What you could to is to writs $$\frac{1}{\log(x)+x}=\frac 1 x \,\,\frac{1}{1+\frac{\log (x)}{x}}=\frac 1 x\sum_{n=0}^\infty (-1)^n \left(\frac{\log (x)}{x}\right)^n=\sum_{n=0}^\infty (-1)^nx^{-(n+1)}\log^n(x)$$ Now $$I_n=\int x^{-(n+1)}\log^n(x)\,dx=\int t^n \, e^{-n t}\, dt=-n^{-(n+1)}\, \Gamma (n+1,n t)$$ $$I_n==-n^{-(n+1)}\, \Gamma (n+1,n \log(x))$$