  We derive an algorithm for computing the total differentials of multi-loop
integrals expressed as one-fold integrals of multiple polylogarithms, which can
involve square roots of polynomials up to degree four and may evaluate to
(elliptic) multiple polylogarithms ((e)MPL). This gives simple algebraic rules
for computing the $(W{-}1, 1)$-coproduct of the resulting weight-$W$ functions
up to period terms, and iterating it gives the symbol without actually
performing any integration. In particular, our algorithm generalizes existing
MPL integration rules and sidesteps the complicated rationalization procedure
in the presence of square roots. We apply our algorithm to conformal
double-$D$-gon integrals in $D$ dimensions with generic kinematics and possibly
massive circumferential propagators. We directly compute, for the first time,
the total differential and symbol (up to period terms) of the $D{=}3$
double-triangle and the $D{=}4$ double-box, which in the special case with
massless propagators represent the first appearance of eMPL functions in
(two-loop) scattering amplitudes of ${\cal N}{=}6$ Chern-Simons-matter theory
and ${\cal N}{=}4$ super-Yang-Mills, respectively.
