I want to approximate the integral of a function over the whole real line very accurately. The integrand should be analytic in a strip but also have exponential decay. A prototypical example is $$\int_{-\infty}^{\infty}\frac{\exp(-x \tanh(x))}{1+x^2},$$ however, I don't know the exact value of this integral and Wolframalpha only yields accuracy to 6 digits (1.49774). Therefore I can't compare my approximation properly.
Is there any closed form of this integral or if not, what would be another good example of an analytic (in a strip) function with exponential decay which has a closed form?