There is a family of pseudocylindrical equal-area projections that starts with the cylindrical equal area and ends with the sinusoidal, parameterised by t ∈ [0,1].
The formulas are:
x = λ cos^t φ
y = ∫_0^φ cos^(1-t) φ
where the integral has no closed form in general.
Update: It turns out this family was described, briefly, by Waldo Tolber in 1973: he calls it “a weighted geometrical mean of the meridians from the Lambert and sinusoidal projections”.