htaccess to remove index.php, with multiple domains - codeigniter

Question

This seems so basic, and there's a plethora of questions regarding .htaccess online, but after two days of research, I still can't make mine work the way I want.
What I want is:
Force https on all requests
Always use the "www" version of the url.
Work on multiple domains (but not redirect them all to a master domain). All my domains point to the same folder (so they'd use the same codebase), and in its root is the .htaccess file.
Remove the "index.php" part of the url, to make it human and SEO friendly.
This is what I have so far:
Start with the basic .htaccess code for CodeIgniter, as shown in the userguide:
RewriteEngine On
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^(.*)$ index.php/$1
Originally, the last line had the [L] flag, but I omitted it, so it will continue to the following rules.
(Am I correct in assuming that it takes the url output in the previous RewriteRule, and perform the following matches on it?)
# for non www urls, add www and force https:
RewriteCond %{HTTP_HOST}(.*) !^www\. [NC]
RewriteRule ^(.*)$ https://www.%{HTTP_HOST}/$1 [R,L]
# for www urls, just force https:
RewriteRule ^(.*)$ https://%{HTTP_HOST}/$1 [R,L]
The above code is achieving tasks 1-3 of my list above, but the index.php is still showing in the address bar.
How do I remove it?

Answer 1

Well, I (partially) gave up on .htaccess, and solved my problem in a different way:
I'm now using CodeIgniter hook to deal with forcing https, and leave htaccess to deal only with forcing www and removing index.php
So I removed the last line (RewriteRule ^(.*)$ https://%{HTTP_HOST}/$1 [R,L]), and added an ssl hook to my application, and all is working now.
The hook function, in case anyone is interested, is this:
function force_ssl()
{
if ($_SERVER['HTTP_HOST']!='localhost' && $_SERVER['HTTP_HOST']!='10.0.2.2')
{
$CI =& get_instance();
$CI->config->config['base_url'] = str_replace('http://', 'https://', $CI->config->config['base_url']);
if ($_SERVER['SERVER_PORT'] != 443) redirect($CI->uri->uri_string());
}
}