export.pl

#!perl
use strict;
use warnings;
use LWP::UserAgent;
use HTTP::Request::Common;
use XML::Atom::Feed;

my $ua = LWP::UserAgent->new;

my $email = 'fill.me@gmail.com';
my $password = 'fillme';

my $res = $ua->request(
    POST 'https://www.google.com/accounts/ClientLogin',
    [
        Email => $email,
        Passwd => $password,
        service => 'writely',
        source => 'test',
        accountType => 'HOSTED_OR_GOOGLE',
    ]
);
my $auth = { split(/[\n=]/, $res->content) }->{Auth} or die "can't get Auth";

my $feed_res = $ua->request(
    GET 'https://docs.google.com/feeds/default/private/full/-/document',
    Authorization => "GoogleLogin auth=\"$auth\"",
    'GData-Version' => '3.0',
);
my $feed = XML::Atom::Feed->new(\($feed_res->content));
my $entry = [ $feed->entries ]->[0] or die "no docs here";
my $url = $entry->content->elem->getAttribute('src');
my $export_res = $ua->request(
    GET $url,
    Authorization => "GoogleLogin auth=\"$auth\"",
    'GData-Version' => '3.0',
);
warn $export_res->as_string;