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I was given this problem in a calculus class many years ago. The teacher didn't have the answer and I am still looking.

$$\int(1+x)^{e^x}dx$$

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    Are you sure you were given $\int (1+x)^{e^x} \mathrm dx$? That one ain't simple...2011-11-06
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    Wolframalpha does not know [the integral](http://www.wolframalpha.com/input/?i=integrate+%281%2Bx%29%5E%28exp%28x%29%29).2011-11-06

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Wolfram Alpha says the following:

http://www.wolframalpha.com/input/?i=integrate+%281%2Bx%29%5Ee%5Ex

In my opinion that integral cannot be expressed by elementary functions.

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According to Maple solution is given by expression:

$x\cdot _2F_1(1,-e^{-x};2;-x)$, where

$_2F_1(1,-e^{-x};2;-x)$ is Hypergeometric_function

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    Are you sure? When I differentiate your purported result, it looks nothing like the OP's integrand...2011-11-06
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    @J.M,Maple output: $x\cdot hypergeom([1,-e^x],[2],-x)$2011-11-06
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    Okay, try differentiating that result you got and see if it's the same as $(1+x)^{e^x}$.2011-11-06
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    Maple 15 does, indeed, say this. But numerically it seems wrong. For the integral from 0 to 1, numerical integration gets 2.482416157 but Maple's antiderivative yields 3.270804242.2011-11-06