 Problem Statement
 Hi, I'm trying to solve the next integral
 Relevant Equations

\begin{equation}
\int ^{\infty} _{0} \dfrac{e^{ax}  e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx
\end{equation}
a > 0 ; b > 0
I split this to get
\begin{equation}
\int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx  \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx
\end{equation}
Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones, but this has, for me, more meaning), but it didn't work: if u=exp(a*x), then
\begin{equation}
\dfrac{1}{a} \int ^{\infty} _{0} \dfrac{1}{(1+u)(1+u^{b/a})} \ dx
\end{equation}
I can't do partial fraction decomposition to this (b/a couldn't be a natural number).
I really don't know how to solve this. I put this into Wolfram Mathematica, but it can't solve it. I forgot mention that this is a multiple choice exercise. The posible answers are: 0 , 1 , ba , (ab) log 2 , [(ab)/ab] log 2.
I suposse that 0 can't be the answer, because the solution of each term on (3) should depende on a and b by the same way (I know, it isn't a good enough reason).
1 neither could be a solution, because the terms in (3) are "simetrics". If the solution of one of then doesn't depend of a or b, then both terms are equal and the solution would be zero.
Then the solution would be some of the other 3 options, but I don't know wich arguments are valid to say wich one.
I hope someone can help me. Sorry for my bad english.
Thanks!
\begin{equation}
\int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx  \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx
\end{equation}
Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones, but this has, for me, more meaning), but it didn't work: if u=exp(a*x), then
\begin{equation}
\dfrac{1}{a} \int ^{\infty} _{0} \dfrac{1}{(1+u)(1+u^{b/a})} \ dx
\end{equation}
I can't do partial fraction decomposition to this (b/a couldn't be a natural number).
I really don't know how to solve this. I put this into Wolfram Mathematica, but it can't solve it. I forgot mention that this is a multiple choice exercise. The posible answers are: 0 , 1 , ba , (ab) log 2 , [(ab)/ab] log 2.
I suposse that 0 can't be the answer, because the solution of each term on (3) should depende on a and b by the same way (I know, it isn't a good enough reason).
1 neither could be a solution, because the terms in (3) are "simetrics". If the solution of one of then doesn't depend of a or b, then both terms are equal and the solution would be zero.
Then the solution would be some of the other 3 options, but I don't know wich arguments are valid to say wich one.
I hope someone can help me. Sorry for my bad english.
Thanks!