I know it might sound like a contradiction, but that is what I feel we are.
Our course of study is rigorous. The manner in which we study our subjects is rigorous. We do not skim over things, we study them in depth. I disagree with the mentality that the lower grades should only focus on the 3 R's. I disagree with the mentality that comprehension in content subjects is irrelevant in the lower grades. I disagree with the mentality that formal education should be held off until age 8 or 10. I homeschool for academic reasons. I homeschool to give my children a rich, well-rounded, challenging academic experience. I don't believe that taking it easy, focusing only on the 3 R's, or delaying formal academics throughout the lower grades is the best way to accomplish my goals. So, I have a rigorous plan for each year. Starting in Kindergarten/1st grade, content subjects utilize a variety of resources ranging from grade level to adult level. We simply don't believe that they should be limited only to materials meant for children of their age. We enjoy using more challenging materials that assist in gaining more depth of knowledge & understanding.
Our daily schedule, though, is rather relaxed. Yes, it is not uncommon for a school day to last 6 hours, and we have a basic outline of what we hope to accomplish each day. However, the kids move at their own pace. I do not set a specific number of hours for the day or set specific start & stop times for subjects. We do not have a strict schedule. I prefer the kids to have a decent amount of say in the day to day workings of our school, because they will one day take over all the day to day time management & organization. I prefer them to work at their own pace, because that will allow for maximum retention & enjoyment.
So, even though it sounds like a contradiction, our homeschooling is both rigorous & relaxed.