I just made up a simple command to do web searches with Cortana. It works just about as good as Google voice search, but the one big advantage it has is its efficiency. Prior to Cortana, I was using Google voice search for browsing the web. It required Vox to open Chrome, pause for x seconds (allow browser to load-- sometimes would not pause long enough because browser took longer to load), then hotkey the voice search. Using Cortana, Vox just inputs the Cortana voice search hotkey and bam, there you go. Super quick.
<?xml version="1.0" encoding="utf-16"?>
<!--VoxCommando 2.1.4.2-->
<groupCollection open="True" name="Cortana Internet Search">
<commandGroup open="True" name="Cortana Internet Search" enabled="True" prefix="" priority="0" requiredProcess="" description="">
<command id="992" name="Hey Cortana" enabled="true" alwaysOn="False" confirm="False" requiredConfidence="0" loop="False" loopDelay="0" loopMax="0" description="">
<action>
<cmdType>InputKeys.Send</cmdType>
<params>
<param>{LWIN}({C})</param>
</params>
<cmdRepeat>1</cmdRepeat>
</action>
<action>
<cmdType>VC.Standby</cmdType>
<params />
<cmdRepeat>1</cmdRepeat>
</action>
<phrase>Interent Search, search, search the internet, web search, search the web, browse the web</phrase>
</command>
</commandGroup>
</groupCollection>