What integrals are known that include the logistic function?

Question

What integrals are known that include the logistic function $\sigma(x):=\frac{1}{1+e^{-x}}$?

I am searching for both definite and indefinite ones as well as approximations!

Answer 1

The logistic function is a common choice of activation function in neural nets. So is the antiderivative$$\int_{-\infty}^x\frac{dt}{1+e^{-t}}=\int_{-\infty}^x\frac{e^tdt}{1+e^t}=\ln(1+e^x),$$the softplus rectifier.

Answer 2

You may want to note that

$$\int_0^\infty \sigma(x) f(x) dx$$

are known as the Complete Fermi-Dirac Integrals. They are typically not solvable but good approximations for $f(x)=x^{k/2}$ with $k\in\mathbb{N}$ have been an active research problem for over a century.

Answer 3

The complete Fermi-Dirac integrals

$$ F_s(x) = \frac{1}{\Gamma(s+1)} \int\limits_{0}^{\infty} \frac{t^s}{e^{t-x}+1} \: dt $$

are related to the polylogarithms, see [http://dlmf.nist.gov/25.12#iii][1]

$$ F_s(x) = -\mathrm{Li}_{s+1}(-e^x). $$

There should be a ton of interesting things here!