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I'm trying to compute the integral $$ \int_{a}^{\infty}\frac{e^x}{x}[\mathrm{Ei}(-x)]^2\,dx, $$ where $\mathrm{Ei}$ is the exponential integral, and $a>0$. The obvious

Integrate[Exp[x]/x*(ExpIntegralE[1, x])^2, {x, a, Infinity}]

doesn't work.

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1 Answer 1

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You will probably only be able to do this numerically.

The function being integrated is

expr = Exp[x]/x*(ExpIntegralE[1, x])^2;

LogPlot[expr, {x, 10^-5, 1}, AxesLabel -> {"x", "expr"}]

enter image description here

Using numeric integration

data = Table[{a, NIntegrate[expr, {x, a, Infinity}]}, 
  {a, 10.^Range[-5, 0, .2]}];

ListLogPlot[data,
 Joined -> True,
 PlotRange -> All,
 AxesLabel -> {"a", "Integral"}]

enter image description here

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