My iphone will not turn on at all, how can I back it up without having to restore it to factory settings?

I dropped my iPhone in water by accident and when I try plugging it into my computer it has a picture of a USB cord and pointing to the iTunes symbol. When I connect it to iTunes it tells me my device is in recovery mode and I either need to update or restore it to factory settings. I've tried updating it and that didn't work, and I've downloaded CopyTrans to backup my iPhone before restoring it and since the screen isn't working, the message to "trust this computer" doesn't appear so the backup app doesn't work. Is there a way I can backup my iPhone without having to "trust this computer"?