Integral with log(1+ix) and complex parameter s

[mmzaj]

We have the integral:
$$I(s)=\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$$
Where $s$ is a complex parameter, and $n$ is a positive integer.
Things i tried:
Set $\log(1+ix)=y$, so that
$$I(s)=-i\int_{c}\log\left(1+\frac{s^{2}}{4\pi^{2}}y^{2} \right )\exp\left(2\pi in e^{y} \right )e^{y}dy$$
Where the path $c$ has to be defined !

 

[soarce]

Inside you have $\log^2$ or $\left| \log \right| ^2$ ?
Have you tried to split the complex logarithm into real and imaginary part ? See for instance http://en.wikipedia.org/wiki/Complex_logarithm