Since I need to use many different input methods at the same time so I wonder if there is a way to switch input method faster than pulling down the notification bar and clicking select input method?
Select Input Method apparently lets you open the IME dialog via a single long press on the Search button. You can use the SIM Test to try this out before paying for the app. However, since most modern phones don't have a Search button, this app has limited usefulness.
Comments suggest that Pie Control can provide a soft Search key which can then be used to activate Select Input Method. However, I found that I was unable to map Search to a "slice of Pie" on my 4.1.2 Galaxy S3 (despite being able to map other controls like Home).
On Android 6 and later you can configure a keyboard change button to appear in the control strip alongside the soft Home and Back buttons whenever input mode is active. But if you have physical Home/Back buttons this probably won't work.
If you are rooted then you can flash a custom ROM such as SlimRom (https://slimroms.org) where you can assign the keyboard change action to long/double tap on Home or on the Back button.