MHB Shortcuts for Solving Multiple Integrals: Is There a Faster Way?

[DaalChawal]

I'm having a problem solving this, My approach is solving $x_1$ as a variable and rest as constants first and then going on further. But it is getting too lengthy. Is there any short method?

 

[Prove It]

Would it help writing the integrand as $\displaystyle \begin{align*} 1 - \frac{2\,x_5}{x_1+x_2+x_3+x_4+x_5} \end{align*}$?

 

[DaalChawal]

Yes, it does help but after the first two steps log comes, and then using by parts it becomes quite lengthy. I was looking for a short approach so that this question can be solved in 2-3 minutes. Btw thanks for your help