How to redirect html extension using htaccess?

At the moment, both of these links show the same page:

http://www.example.com/podcast/episode.html
http://www.example.com/podcast/episode

What I want to do is redirect all the html-type links to the non-html links.

I know this might seem simple to do with htaccess, but it's not working for me.

Here's my htaccess code so far:

    RewriteEngine On
    RewriteBase /

    #removing trailing slash

    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteRule ^(.*)/$ $1 [R=301,L]

    #non www to www

    RewriteCond %{HTTP_HOST} !^www\.
    RewriteRule ^(.*)$ http://www.%{HTTP_HOST}/$1 [R=301,L]

    #shtml

    AddType text/html .html
    AddHandler server-parsed .html

    #html

    RewriteCond %{REQUEST_FILENAME} !-f
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteRule ^([^\.]+)$ $1.html [NC,L]

    #index redirect

    RewriteCond %{THE_REQUEST} ^[A-Z]{3,9}\ /index\.html\ HTTP/ 
    RewriteRule ^index\.html$ http://example.com/ [R=301,L]

RewriteCond %{THE_REQUEST} \.html
RewriteRule ^(.*)\.html$ /$1 [R=301,L]