How can the indefinite integral of e^(1/x)/(x(x+1)^2) be solved by hand?

[mishima]

$$\int \frac {e^{1/x}} {x(x+1)^2} \, dx$$

I came across this indefinite integral when solving a second order differential equation using reduction of order. My CAS can solve it easy enough, but I was wondering what technique could be used to solve it by hand. I have tried some standard approaches without much luck. Thanks for any insights.

 

[member 587159]

The first step is to substitute $u = 1/x$ and after that substitution integration by parts.

Here is a site where you can enter the integral and it will give you a worked-out solution: https://www.integral-calculator.com/

 

[mishima]

Thanks, I was making a mistake with integration by parts. Nifty site!