Friendly challenge - alternate solutions for simplifying $\int \frac{x+1}{x-1} \, \mathrm dx$

Question

I have a habit of more or less "making up" problems and trying to solve them in my free time. This particular one I found to be fun, and thought others might enjoy trying to solve it using other means. Have fun, enjoy and feel free to post your solutions/suggestions.

Here is my solution:
$\int \frac{x+1}{x-1} \, dx$ =
$\int \frac{x}{x-1} + \frac{1}{x-1} \,dx$ =
$\int \frac{1}{x-1} + \frac{1}{x-1} +1 \,dx$ =
$\int \frac{2}{x-1} + 1 \,dx$ =
$2ln|x-1|+x+C $

Answer 1

$$\int\dfrac{x+1}{x-1}dx=\int\left(1+\dfrac2{x-1}\right)dx=x+2\ln\mid x-1\mid+\,C$$