Script timeout when printing invoices with more than 5 line items

When the 'Tax and Invoicing' component of CiviContribute is enabled, each contribution allows the user to 'print invoice'. When a contribution includes multiple line items, for example, when an event registration includes multiple participants, the 'print invoice' process will never complete and timeout.

In our case, the Nginx error log shows this:

2016/02/06 15:03:58 [error] 1039#0: *1775 upstream timed out (110: Connection timed out) while reading response header from upstream, client: <ip address>, server: <server_name>, request: "GET /civicrm/contribute/invoice?reset=1&id=207&cid=11628 HTTP/1.1", upstream: "fastcgi://unix:/var/run/php5-fpm.sock", host: "<hostname>", referrer: "<site url>/civicrm/contact/view/contribution?reset=1&id=207&cid=11628&action=view&context=search&selectedChild=contribute"

We've increased Nginx and FPM execution time settings, we've optimized MYSQL, installed memcached and optimized it. Nothing has worked. The script that produces the invoice PDF in this context is just too slow when the invoice includes more than 5 line items or so.