It's unlikely that you would ever use Ritual Spellcasting with Increase Reflexes. Ritual Spellcasting is used for rituals (e.g., Curse, Prodigal Spell, Remote Sensing, Ward, Circle of Protection, Circle of Healing, Renascence, Watcher, and Homonculus from the core rulebook, p.296-299).
Say you want to ward your apartment against astral visitors. You learn the Ward ritual, pick up some reagents from your talismonger, and set to work. You could do this with a group, but it works by yourself too. Following the steps for ritual magic (p.295-296), you choose the ritual leader (yourself), the ritual (Ward), and the Force (let's say 4). You set up the foundation (your own lodge, at least rating 4), give the offering (4 reagents), and perform the ritual (4 hours, presumably walking the perimeter and marking the warded surfaces with symbols meaningful to you and your tradition). Finally, you seal the ritual by rolling Ritual Spellcasting + Magic [Force 4] vs. (Force x 2 = 8 dice). You get two net hits, so according to the ritual description, your ward will last two weeks. Or you could spend 4 karma and make it permanent. Then you resist drain, equal to twice the number of hits from the opposing (Force x 2) test.
Some rituals are designed to enhance regular spells (e.g., Curse for Illusions, Prodigal Spell for Combat, Remote Sensing for Detection, Circle of Healing for Health). Say you want to get some petty revenge against that Knight Errant goon who spat on you this morning, and you prepare a Curse ritual (p.297) to hit him with a Bugs illusion spell (p.290). You cast Preserve (SG p.117) on the loogie earlier, so you have the necessary Material Link. Again, following the steps for ritual magic: you choose the leader (yourself), the ritual (Curse), and the Force (4 again, since you're casting alone). You set the foundation (your lodge), give the offering (4 reagents), and perform the ritual (in this case, you cast the Bugs spell with Spellcasting + Magic, and resist the drain, although it takes 4 hours instead of a Complex action). Finally, you seal the ritual with Ritual Spellcasting + Magic vs. (Force x 2) and resist the drain. Your Bugs spell zooms through astral space following the path created by the Material Link, and strikes the hapless KE goon just as he sits back to watch the evening Combat Biker game on his trideo. (He gets to make the usual Willpower + Logic resistance roll vs. your Bugs spellcasting test, but fails.) Illusory Bugs swarm over him for as long as you feel like sustaining the spell; but bear in mind that if the goon has a magician buddy nearby, they can track the spell back to you while you sustain it. That works two ways: If you want to know where the goon lives, you can track the spell to him.
