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Is it possible to evaluate the following integral? $$ \int_0^{\tau} \frac{\log{\left( \sum_i^n a_i e^{-s_i t} \right)}}{t} dt $$ where $a_i$ and $s_i$ are sets of constants:

$0 \leq a_i \leq 1 \ \forall i$ and $\sum_i a_i = 1$.

$0 \leq s_i \ \forall i$.

$n$ is a finite integer $> 1$.

Thank you!

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    is that a finite or infinite sum? – 2017-01-24
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    @GFauxPas The sum is finite. I've edited the notation in the question to reflect that. – 2017-01-24
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    I don't think this integral converges. – 2017-01-24
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    Putting aside the closed form, the integral is finite only when $\sum_{i=1}^{n} a_i = 1$. – 2017-01-24
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    @SangchulLee It is in fact the case that $\sum_i a_i = 1$ for my application. Do you have any idea how to approach evaluating the integral in closed form? – 2017-01-25
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    I am afraid that I have no idea. Except for the simplest case $n = 1$ with a trivial answer, even the case $n = 2$ produces an integral with no elementary closed form. But thinking that the the denominator reminds me of a partition function, I humbly suspect that the exact formula is possibly not exactly what you want to read out from the integral. – 2017-01-25

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