Once Anthy is installed, open the Fcitx configuration application by typing "input" on the search field of the KickOff menu:
Once there, deactivate the box "Only show current language." This action will show all the available languages. Locate entries for "Japanese" and select Anthy. Click on the arrow pointing to the right to add Anthy as an alternative input method.
Then click "Apply" and "OK" to leave. That is all.
On the task bar, locate the keyboard icon (Fcitx) and right click on it. Follow the menu to select "Input Method" and select Anthy.
You will see that the keyboard icon changes to Anthy's icon.
Now you can type in Japanese on LibreOffice Writer. :)