BusyBox combines tiny versions of many common UNIX utilities into a single small executable. It provides minimalist replacements for most of the utilities you usually find in GNU coreutils, util-linux, etc. The utilities in BusyBox generally have fewer options than their full-featured GNU cousins; however, the options that are included provide the expected functionality and behave very much like their GNU counterparts.
Now, how do we do installing this so called busybox? The first method is using busybox zip installer if you don't have root access but already have custom recovery. The second option is using busybox app installer if you already gain a root access but don't have custom recovery installed. This is the most preferably way to install busybox binary. The most popular busybox installer used is Busybox Installer which can easy to get from Google Playstore.
Both methods mentioned above are not installing systemlessly. This article will show you how to install any busybox binary in your android without messing you system partition. This way you will have a lasting busybox even when you changing your rom as long as you using SuperSu as root access manager.
Yep, whatever version of precompiled busybox binary you used, just install it along with SuperSU image. Don't get it? I'll explaon how SuperSU works systemlessly in flash. SuperSU make their own partition image (named with su.img) placed in /data partition. Then SuperSu place SU binary in this image. They modified the boot partitions to make it Mount su.img as /su not inside the /system.
So if you want to install a ststemless busybox binary, then install it after you have SuperSU and place your busybox binary at /su/xbin not in /system/xbin. For some busybox app detectors may not detecting a busybox on your system even it's already installed. But don't worry, you still have it functionality. Not believing? Then try to type
from your terminal apps.
Have a complaint? Don't hesitate to write down in comments section. Cayoo..