Another indefinite integral I need to perform

Question

In my work, I need to solve a third order linear ODE, which comes from linearizing a nonlinear PDE (in space $x$ and time $t$) about a traveling wave localized solution (think of a solitary wave, a soliton, or a traveling front). The linear ode is obtained by assuming separation of variables of the form $w=e^{\lambda t}f(\xi)$ ($\xi=x-ct$ is the traveling waves variable) for the solution of the linear PDE resulting from the linearization. The idea is that if the linear PDE has a solution of the form $w=e^{\lambda t}f(\xi)$ with $\Re({\lambda})>0$ and with $f\in L^2(\mathbb{R})$, then the localized solution is linearly unstable. Since you assume separation of variables in the linearization, you obtain an eigenvalue problem of the form $\mathcal{L}f=\lambda f$, where $\mathcal{L}$ is a differential operator in $\xi$, which happens (in my case) to be third oder. Usually one cannot find solutions to such a linear ODE since it is non-autonomous. However, exceptionally in my case, I was able to find two explicit solutions. I applied reduction of order and, among the integrals I need to perform to find a third linearly independent solution, there is this one below. I would like to know as much as I can about that antiderivative. In particular, I would like to know if there is an explicit function I can use (for example in Maple) to manipulate the corresponding solution. In the end, the issue is that I am confronted with the following indefinite integral. I would appreciate any help finding it explicitly. $$ \int{\frac{e^{bx}}{e^x-1}\,{\rm{d}}x}, $$ where $b$ is a real number.

Answer 1

$$ \begin{align} I=&\int{\frac{e^{bx}}{e^x-1}\,{\rm{d}}x}\\ =&\int{\frac{e^{(b-1)x}}{1-e^{-x}}\,{\rm{d}}x}\\ =&\int{e^{(b-1)x}\sum_\limits{n=0}^{\infty}e^{-nx}\,{\rm{d}}x}\\ =&\int {\sum_\limits{n=1}^{\infty}e^{(b-n)x}\,{\rm{d}}x}\\ =&\sum_\limits{n=1}^{\infty}\frac {e^{(b-n)x}}{b-n}\\ \end{align} $$