  This paper presents a new uniquely solvable boundary integral equation for
computing the conformal mapping, its derivative and its inverse from bounded
multiply connected regions onto the five classical canonical slit regions. The
integral equation is derived by reformulating the conformal mapping as an
adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we
derive a boundary integral equation with the adjoint generalized Neumann kernel
for the derivative of the boundary correspondence function $\theta'$. Only the
right-hand side of the integral equation is different from a canonical region
to another. The function $\theta'$ is integrated to obtain the boundary
correspondence function $\theta$. The integration constants as well as the
parameters of the canonical region are computed using the same uniquely
solvable integral equation.
  A numerical example is presented to illustrate the accuracy of the proposed
method.
