\sum_i
    \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]
  d_{ni} 
  \left[
    A e^{-\imath \omega_0 t} + A^* e^{+\imath \omega_0 t} 
  \right]
  \exp \left[ -\imath W_{in} t \right]
\\
&=&
-\frac{1}{ \hbar}
\sum_i
    \sum_j
      c_j
      d_{ni} 
      d_{ij} 
  \left[ 
    A
    \frac{
          \exp \left[ -\imath \left( W_{jn} + \omega_0\right) t \right]
         }
         {W_{ij}-\omega_0}
    +
    A^*
    \frac{
          \exp \left[ -\imath \left( W_{jn} - \omega_0 \right) t \right]
         }
         {W_{ij}+\omega_0}
  \right]
  \left[
    A e^{-\imath \omega_0 t} + A^* e^{+\imath \omega_0 t} 
  \right]
\\
&=&
-\frac{1}{ \hbar}
\sum_i
    \sum_j
      c_j
      d_{ni} 
      d_{ij} 
  \left\{
    A^2
    \frac{
          \exp \left[ -\imath \left( W_{jn} + 2 \omega_0\right) t \right]
         }
         {W_{ij}-\omega_0}
   +
    A A^*
    \frac{
          \exp \left[ -\imath W_{jn} t \right]
         }
         {W_{ij}-\omega_0}
\right.
\nonumber
\\
&& ~~~~ ~~~~ ~~~~ ~~~~ ~~~~ ~~~~ ~~~~ ~~~~ 
\left.
   +
    A^* A 
    \frac{
          \exp \left[ -\imath W_{jn} t \right]
         }
         {W_{ij}+\omega_0}
   +
    {A^*}^2
    \frac{
          \exp \left[ -\imath \left( W_{jn} - 2 \omega_0 \right) t \right]
         }
         {W_{ij}+\omega_0}
  \right\}
\\
&\approx&
-\frac{1}{ \hbar}
\sum_i
    \sum_j
      c_j
      d_{ni} 
      d_{ij} 
      A A^*
          \exp \left[ -\imath W_{jn} t \right]
  \left\{
    \frac{1}
         {W_{ij}-\omega_0}
   