There are two types of boot animation:
I was able to change ROM-based animation by rooting the phone and using a custom ROM.
However, I'm a bit confused about how to change or remove the brand logo animation. Is it safe to do that? Will it increase the speed of the booting process?
I have a very different take on this matter than @sbrm1. Note a few things first and then decide whether doing this is worth it:
The boot logo is a actually a raw binary image, constrained by size, not format.
The format of the bootloader splash screen, as it is known, varies for even models of the same device (case in point, Exynos and Qualcomm Samsung Galaxy models).
On Qualcomm devices, the bootloader splash logo is typically stored on a partition called splash or logo, to change which you need to flash as similarly named .mbn or .bin file (not .img)
If you don't choose a logo with a format and size specific to your device, it may result in unforeseen circumstances (read, what will appear to be a Hard Brick, the device not booting up at all). Why? The bootloader is dumb and will try to load whatever is on the splash partition as long as it fits in there (i.e., has a specific size). It might not even be a picture at all. The bootloader may choose to 'panic', i.e., show some error message and reboot, keep trying to reload the splash, or continue booting the OS. You can hope for the last, but you never know unless someone has documented the specific behaviour for your device or you learn it the hard way. You may or may not lose your charging screen, however.
Do you still want to do this? I don't know of a guide specific to your device, let that put you off a bit. If there is indeed an acceptable splash file for your device, just fastboot flash the image file (note, image as in disk image, not photograph) to whichever of the above 2 mentioned splash partitions which exists on your device. This can be done on some HTC, LG and Nexus devices with Qualcomm or Tegra chipsets.
A last FYI, the manufacturer may choose to integrate the logo into the bootloader (aboot). I know Xiaomi does this for some devices (Redmi 1s and Redmi 2 - personal experience), it's probably true in your case too. Then you will need a custom bootloader, with the splash screen packed into the bootloader flashable image file (image as above).
Still there? I can't help you beyond discouraging you, so...
Q & A