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odd sine
save one sine, convert the rest to cosines
odd cosine
save one cosine, convert the rest to sines
even both
use half angle identities, convert to odd powers of cosine
even secant
save a secant^2, convert the rest to tangents
odd tangent
save a secant-tangent, convert the rest to secants
even tangent, no secants
convert a tangent^2 to a secant^2, expand and repeat
odd secant, no tangents
integration by parts