I have an integral
$$I(a,b,c,d) = \int_{-\infty}^0 \bigg(\tanh(ax+b)\tanh(cx+d) - 1\bigg)~\mathrm{d}x$$
which I need to approximate analytically (to use it in further steps within a machine learning approach) with $a,c\in\mathbb{R}_+$ and $b,d\in\mathbb{R}$. What would be the best approach to obtain an anaytic expression approximating $I(a,b,c,d)$?
For $a=c$, I get an answer
$$I(a,b,c,d) = -\frac{1}{a}(\coth(d - b) (d + \log(\cosh(d)) + \log(1 - \tanh(b))))$$
unless I screwed up? I used the definition of $\tanh(\cdot)$, rewrote it in terms of $p=\exp(ax)$ and did partial fraction decomposition.