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For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$:

$$\int_{\delta}^{1}\frac{(a^{x} -1)(a^{1-x} - 1)}{x(1-x)}dx, 0 < \delta < 1$$

My friend has informed me that she has applied the Risch algorithm to this integral and it is not expressible in terms of elementary functions. I believe you can express this integral in terms of exponential integrals (and numerically approximate using Ramanujan's fast convergence formula for the exponential integral), but I'm not sure how to deal with negative real values of the exponential integral if this method is useful. Are there any other alternative ideas for how I may compute this integral?

Any help or suggestions would be greatly appreciated!

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    $\begingroup$ yes, it's an Exponential Integral, just feeding it to Mathematica (for a numerical answer) does not seem to give it problems. $\endgroup$
    – Carlo Beenakker
    Commented Feb 3, 2015 at 20:48
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    $\begingroup$ I'd like to know how to approximate the exponential integral of a negative value? $\endgroup$
    – Samuel Reid
    Commented Feb 3, 2015 at 21:55
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    $\begingroup$ Actually Mathematica says it is$$\operatorname{Ei}((1-\delta)\log(a)) + a \operatorname{Ei}(-(1- \delta)\log(a))- (\operatorname{Ei}(\delta\log(a)) + a \operatorname{Ei}(-\delta\log(a))) -2 (1 + a)\operatorname{arctanh}(1 - 2\delta)$$and the Wolfram functions site has the transformation rules for $\operatorname{Ei}(\sqrt{z^2})$ (see functions.wolfram.com/GammaBetaErf/ExpIntegralEi/16/01/01) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 3, 2015 at 21:55

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