  We prove mixed inequalities for commutators of Calder\'on-Zygmund operators
(CZO) with multilinear symbols. Concretely, let $m\in\mathbb{N}$ and
$\mathbf{b}=(b_1,b_2,\dots, b_m)$ be a vectorial symbol such that each
component $b_i\in \mathrm{Osc}_{\mathrm{exp}\, L^{r_i}}$, with $r_i\geq 1$. If
$u\in A_1$ and $v\in A_\infty(u)$ we prove that the inequality
  \[uv\left(\left\{x\in \mathbb{R}^n:
\frac{|T_\mathbf{b}(fv)(x)|}{v(x)}>t\right\}\right)\leq
C\int_{\mathbb{R}^n}\Phi\left(\|\mathbf{b}\|\frac{|f(x)|}{t}\right)u(x)v(x)\,dx\]
holds for every $t>0$, where $\Phi(t)=t(1+\log^+t)^r$, with $1/r=\sum_{i=1}^m
1/r_i$.
  We also consider operators of convolution type with kernels satisfying less
regularity properties than CZO. In this setting, we give a Coifman type
inequality for the associated commutators with multilinear symbol. This result
allows us to deduce the $L^p(w)$-boundedness of these operators when
$1<p<\infty$ and $w\in A_p$. As a consequence, we can obtain the desired mixed
inequality in this context.
