Remember that structured anapanasati practice is the highest expression of Satipatthana:
"And how is mindfulness of in-&-out breathing developed & pursued so as to bring the four frames of reference to their culmination?
"[1] On whatever occasion a monk breathing in long discerns, 'I am breathing in long'; or breathing out long, discerns, 'I am breathing out long'; or breathing in short, discerns, 'I am breathing in short'; or breathing out short, discerns, 'I am breathing out short'; trains himself, 'I will breathe in...&... out sensitive to the entire body'; trains himself, 'I will breathe in...&...out calming bodily fabrication': On that occasion the monk remains focused on the body in & of itself — ardent, alert, & mindful — putting aside greed & distress with reference to the world. I tell you, monks, that this — the in-&-out breath — is classed as a body among bodies, which is why the monk on that occasion remains focused on the body in & of itself — ardent, alert, & mindful — putting aside greed & distress with reference to the world.
"[2] On whatever occasion a monk trains himself, 'I will breathe in...&...out sensitive to rapture'; trains himself, 'I will breathe in...&...out sensitive to pleasure'; trains himself, 'I will breathe in...&...out sensitive to mental fabrication'; trains himself, 'I will breathe in...&...out calming mental fabrication': On that occasion the monk remains focused on feelings in & of themselves — ardent, alert, & mindful — putting aside greed & distress with reference to the world. I tell you, monks, that this — careful attention to in-&-out breaths — is classed as a feeling among feelings,[6] which is why the monk on that occasion remains focused on feelings in & of themselves — ardent, alert, & mindful — putting aside greed & distress with reference to the world.
"[3] On whatever occasion a monk trains himself, 'I will breathe in...&...out sensitive to the mind'; trains himself, 'I will breathe in...&...out satisfying the mind'; trains himself, 'I will breathe in...&...out steadying the mind'; trains himself, 'I will breathe in...&...out releasing the mind': On that occasion the monk remains focused on the mind in & of itself — ardent, alert, & mindful — putting aside greed & distress with reference to the world. I don't say that there is mindfulness of in-&-out breathing in one of lapsed mindfulness and no alertness, which is why the monk on that occasion remains focused on the mind in & of itself — ardent, alert, & mindful — putting aside greed & distress with reference to the world.
"[4] On whatever occasion a monk trains himself, 'I will breathe in...&...out focusing on inconstancy'; trains himself, 'I will breathe in...&...out focusing on dispassion'; trains himself, 'I will breathe in...&...out focusing on cessation'; trains himself, 'I will breathe in...&...out focusing on relinquishment': On that occasion the monk remains focused on mental qualities in & of themselves — ardent, alert, & mindful — putting aside greed & distress with reference to the world. He who sees with discernment the abandoning of greed & distress is one who watches carefully with equanimity, which is why the monk on that occasion remains focused on mental qualities in & of themselves — ardent, alert, & mindful — putting aside greed & distress with reference to the world.
"This is how mindfulness of in-&-out breathing is developed & pursued so as to bring the four frames of reference to their culmination.
Anapanasati is the Buddha's Jhana, which of course develops samadhi and sati/samatha and vipassana together. So you can enter the first Jhana at the end of the first tetrad (when pitisukha begins to arise) and then proceed all the way to the fourth Jhana by the time you hit the final tetrad, where you can do the hardcore insight stuff "permeating the body with a pure, bright awareness." By the time you've
completely and totally calmed the body, calmed piti and sukha, and completely gladdened/liberated the mind, it's hard to believe you wouldn't be in the fourth Jhana. But obviously you can also do a lighter version in the first Jhana, or an even lighter version without any form of Jhana.
Either way, the bottom line is that anapanasati is the most perfect form of satipatthana meditation, and it develops both samatha and vipassana together, the first three tetrads being primarily samatha supported by vipassana and the last tetrad being vipassana supported by samatha. Either way, you have both, and both can be perfected in Jhana.
Remember the Dhammapada:
There's no jhana for one with no discernment,
no discernment for one with no jhana.
But one with both jhana and discernment:
he's on the verge of Unbinding.