# Integral of a function in the exponential

I would like to calculate the following integral:

$$\int \exp\left[a \frac{1-e^{-\kappa_1 s}}{\kappa_1}+b\frac{1-e^{-\kappa_2 s}}{\kappa_2}+c\times s\right]ds$$

This is how I proceeded:

Let's define $u\triangleq e^{c\times s}$! Then we have:

$du=c\times u\times ds$

where $a, b, c, \kappa_1$ and $\kappa_2$ are constants.

Thus the original integral is:

$\int u \exp\left[\frac{a}{\kappa_1}+\frac{b}{\kappa_2}\right]\exp\left[-\frac{a}{\kappa_1}e^{-\kappa_1 s}-\frac{b}{\kappa_2}e^{-\kappa_2 s}\right]ds \\ =\frac{1}{c}\exp\left[\frac{a}{\kappa_1}+\frac{b}{\kappa_2}\right]\int \exp\left[-\frac{a}{\kappa_1}u^{-\frac{\kappa_1}{c}}-\frac{b}{\kappa_2}u^{-\frac{\kappa_2}{c}}\right]du$

But from here I could not go any further. Any hints and help would be greatly appreciated!

• Have you any reason to think that a closed form could exist for the antiderivative ? I guess that, for a small range of integration, we could approximate it. If this is the case, just post. – Claude Leibovici Jun 7 '15 at 2:15

\begin{align} \int e^{a\frac{1-e^{-\kappa_1s}}{\kappa_1}+b\frac{1-e^{-\kappa_2s}}{\kappa_2}+cs}\ \text{d}s & = e^{\frac{a}{\kappa_1}+\frac{b}{\kappa_2}}\int e^{-\frac{ae^{-\kappa_1s}}{\kappa_1}-\frac{be^{-\kappa_2s}}{\kappa_2}}e^{cs}\ \text{d}s \\ & = e^{\frac{a}{\kappa_1}+\frac{b}{\kappa_2}}\int\sum\limits_{n=0}^\infty\dfrac{(-1)^n\left(\dfrac{ae^{-\kappa_1s}}{\kappa_1}+\dfrac{be^{-\kappa_2s}}{\kappa_2}\right)^ne^{cs}}{n!}\ \text{d}s \\ & = e^{\frac{a}{\kappa_1}+\frac{b}{\kappa_2}}\int\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^nC_k^n\dfrac{a^ke^{-\kappa_1ks}b^{n-k}e^{-\kappa_2(n-k)s}}{\kappa_1^k\kappa_2^{n-k}}e^{cs}}{n!}\ \text{d}s \\ & = e^{\frac{a}{\kappa_1}+\frac{b}{\kappa_2}}\int\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^na^kb^{n-k}e^{(c-\kappa_1k-\kappa_2(n-k))s}}{k!(n-k)!\kappa_1^k\kappa_2^{n-k}}\ \text{d}s \\ & = \sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^na^kb^{n-k}e^{(c-\kappa_1k-\kappa_2(n-k))s+\frac{a}{\kappa_1}+\frac{b}{\kappa_2}}}{k!(n-k)!\kappa_1^k\kappa_2^{n-k}(c-\kappa_1k-\kappa_2(n-k))}+C \\ & = \sum\limits_{k=0}^\infty\sum\limits_{n=k}^\infty\dfrac{(-1)^na^kb^{n-k}e^{(c-\kappa_1k-\kappa_2(n-k))s+\frac{a}{\kappa_1}+\frac{b}{\kappa_2}}}{k!(n-k)!\kappa_1^k\kappa_2^{n-k}(c-\kappa_1k-\kappa_2(n-k))}+C \\ & = \sum\limits_{k=0}^\infty\sum\limits_{n=0}^\infty\dfrac{(-1)^{n+k}a^kb^ne^{(c-\kappa_1k-\kappa_2n)s+\frac{a}{\kappa_1}+\frac{b}{\kappa_2}}}{k!n!\kappa_1^k\kappa_2^n(c-\kappa_1k-\kappa_2n)}+C \end{align}