Solve the following definite integral

Question

This integral was given in an electrochemistry textbook and I am having trouble with it.$${-RT\over F}\sum_i\int_{c_i(a)}^{c_i(b)}{t_i\over z_i}d ln(c_i)$$Here R, T and F are constants (gas constant, temperature and Faradays constant). $c_i$ goes from ${c_i(a)}$ to ${c_i(b)}$. $z_i$ is a constant for a given i. And,$$t_i = {|z_i|c_iu_i\over \sum_i|z_i|c_iu_i}$$Where $z_i$ and $c_i$ are the same as above and $u_i$ is constant for a given i. According to the textbook the expression becomes$${\sum_i{|Z_i| \over z_i}u_i[c_i(b)-c_i(a)]\over \sum_i|z_i|u_i[c_i(b)-c_i(a)]}{RT\over F}{ln\left({\sum_iz_iu_ic_i(a)\over \sum_iz_iu_ic_i(b)}\right)}$$But my attempt got me stuck as follows:$${-RT\over F}\sum_i \int_a^b{{|z_i|\over zi}u_i\over \sum_i|z_i|c_iu_i}dc_i$$The integral is of the form$\int{1\over {ax + b}}dx$ where $a = |z_i|u_i$ and $b = \sum_{j \ne i}|z_j|c_ju_j$ $${-RT\over F}\sum_i{1\over z_i}ln\left( {|z_i|u_ic_i(b)+\sum_{j \ne i}|z_j|c_ju_j\over |z_i|u_ic_i(a)+\sum_{j \ne i}|z_j|c_ju_j}\right)$$At which point, I'm stuck and I can't get to the answer that was given in the textbook. What am I missing. I'm hoping that no additional chemistry relations are required to solve reach the equation