# Logistic curve integral

I am trying to solve the integral of the following equation:

y= a + (b/(1+(x/c)^d)

with a,b,c,d as constant

I will apprecciate your help

Thanks

## Answers and Replies

This indefinite integral cannot be expressed with a combination of a finite number of elementary functions. It requires special functions, namely the Gaussian Hypergeometric function (2F1), or another special function of lower level namely the incomplete Beta function (but in complex domain).

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This indefinite integral cannot be expressed with a combination of a finite number of elementary functions. It requires special functions, namely the Gaussian Hypergeometric function (2F1), or another special function of lower level namely the incomplete Beta function (but in complex domain).

Thank you Jacqueline,
I am trying to calculate the area under the curve but I do not know how to arrange the Gaussian Hypergeometric function or the Beta function.

Can you help me, please? Can you explain the procedure to reach my goal?

For example:
a= 2,43; b= 6; c= 2.5; δ= 9.68 with interval: x1=2.5 and x2=7.5

it is not a question of result, but I would like to understand the porcedure.

Thank you in advance

It is possible to compute the numerical value of an hypergeometric function "by hand", thanks to series expansion. But it would be awfully boring and time consuming. That was the way to do it in the good old days, before the age of computers.
If you have "Mathematica" or any other math. package, certainly, the hypergeometric function is included, allowing to compute numerical values.
But the most simple way is to use a numerical integrator. Even some pocket calculator can do that. Just program the function to integrate (no need for hypergeometric function). Also more advanced softwares are very simple to use, for example MatCad (screen copy below)
Some calculators are on free access on the web., for example WolframAlpha (screen copy below), use Google to find them.