~} &c_n, d_{ni}&
\textrm{vary only slowly, so I can integrate just the exponentials }
\\
c_i
&=&
-\frac{1}{ \imath \hbar}
\sum_j
  c_j
  d_{ij} 
  \left[ 
    A
    \frac{
          \exp \left[ -\imath \left( W_j - W_i + \omega_0\right) t \right]
         }
         {-\imath \left( W_j-W_i+\omega_0 \right) }
    +
    A^*
    \frac{
          \exp \left[ -\imath \left( W_j - W_i - \omega_0 \right) t \right]
         }
         {-\imath \left( W_j-W_i-\omega_0 \right) }
  \right]
\\
&=&
\frac{1}{ \hbar}
\sum_j
  c_j
  d_{ij}
  \left[ 
    A
    \frac{
          \exp \left[ -\imath \left( W_j - W_i + \omega_0\right) t \right]
         }
         {W_i-W_j-\omega_0}
    +
    A^*
    \frac{
          \exp \left[ -\imath \left( W_j - W_i - \omega_0 \right) t \right]
         }
         {W_i-W_j+\omega_0}
  \right]\end{aligned}$$

Note the swap of $\left(W_j-W_i\right)$ to $-\left( W_i-W_j \right)$ in the denominators. Now substitute the $c_i$ solution into the $d c_n / dt$ equations (using $l\rightarrow j$), and introduce the shorthand notation $W_{ij} = W_i - W_j$, $$\begin{aligned}
\imath \hbar \frac{d c_n}{dt}
&=&
-
\sum_i
  \left\{
    \frac{1}{ \hbar}
    \sum_j
      c_j
      d_{ij} 
  \left[ 
    A
    \frac{
          \exp \left[ -\imath \left( W_{ji} + \omega_0\right) t \right]
         }
         {W_{ij}-\omega_0}
    +
    A^*
    \frac{
          \exp \left[ -\imath \left( W_{ji} - \omega_0 \right) t \right]
         }
         {W_{ij}+\omega_0}
  \right]
  \right\}
  d_{ni} E 
  \exp \left[ -\imath W_{in} t \right]
\\
&=&
-\frac{1}{ \hbar}
