How would we find $ \int e ^{x(e^x+1)} dx $ ?
If the integral cannot be expressed in elementary functions, then how can we prove that it cannot be expressed in elementary functions?
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This might be in your interest: https://math.stackexchange.com/q/265780/515527 and this: https://math.stackexchange.com/q/155/515527 – Zacky Feb 09 '19 at 11:37
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By using $e^{x}=u , e^{x}dx=udx=du $ we reach to this one: $$ \int e^{x(e^{x}+1)}dx=\int e^{xe^{x}}.e^{x}dx=\int e^{xu}.u\frac{du}{u}=\int (e^{x})^{u}du=\int u^{u}du$$
I don't continue for integration of this equation. Please refer to these pages:
BarzanHayati
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