Can anyone solve this integral?

Question

I want to know what is the integral of $$y=\frac{\exp(a+bx+cx^2)}{1+\exp(a+bx+cx^2)}.$$

Thanks so much. I can remember how to integrate.

Answer 1

Assume $c\neq0$ for the key case:

\begin{align} \int \frac {e^{a+bx+cx^2}} {1+e^{a+bx+cx^2}} \,\Bbb dx &= \int \frac 1 {e^{-a-bx-cx^2}+1} \,\Bbb dx \\ &= \int \frac 1 {e^{-c\left(x^2+\frac{bx}{c}+\frac{a}{c}\right)}+1} \,\Bbb dx \\ &= \int \frac 1 {e^{-c\left(x^2+\frac{bx}{c}+\frac{b^2}{4c^2}+\frac{a}{c}-\frac{b^2}{4c^2}\right)}+1} \,\Bbb dx \\ &= \int \frac 1 {e^{-c\left(\left(x+\frac{b}{2c}\right)^2+\frac{4ac-b^2}{4c^2}\right)}+1} \,\Bbb dx \\ &= \int \frac 1 {e^\frac{b^2-4ac}{4c}e^{-c\left(x+\frac{b}{2c}\right)^2}+1} \,\Bbb dx \\ \left(\text{Let }u=x+\frac{b}{2c}\right) \\ &= \int \frac 1 {e^\frac{b^2-4ac}{4c}e^{-cu^2}+1} \,\Bbb du \\ (\text{Let }v=u^2) \\ &= \int \frac 1 {2\sqrt{v}\left(e^\frac{b^2-4ac}{4c}e^{-cv}+1\right)} \,\Bbb dv \end{align}

Which may relate to the incomplete polylogarithm function.