Creating a simple HTTPS redirect rule for Apache can potentially be a long, tiresome journey of trial and error.
There are many ways to write the rule with subtle syntax differences and other factors such as the version of the Apache server also affecting the outcome.
The following snippet is what finally worked for me on Gigahost (a shared web hosting provider in Denmark).

<IfModule mod_rewrite.c>
    RewriteEngine On
    RewriteCond %{ENV:HTTPS} !on
    RewriteRule ^(.*)$ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]
</IfModule>