Integral of $e^{x^{-1}+x^2}$

Question

The integral appears to be non-elementary. I have tried integration by parts using the exponential integral and the error function, but came up empty. Just wondering if anyone has any ideas on how to attack this. Thanks in advance.

Answer 1

There's not a whole lot about this that's elementary. I guess you can say \begin{align*} e^{x^{-1}+x^2}&=\sum_{k=0}^\infty \frac{(x^{-1}+x^2)^k}{k!} \\ &=\sum_{k=0}^\infty \frac{1}{k!}\sum_{j=0}^k {k \choose j}x^{-j}x^{2(k-j)} \\ &=\sum_{k=0}^\infty \frac{1}{k!}\sum_{j=0}^k {k \choose j}x^{2k-3j} \end{align*} so the antiderivative is $$ \sum_{k=0}^\infty \frac{1}{k!}\sum_{j=0}^k {k \choose j}\frac{x^{2k-3j+1}}{2k-3j+1} + C. $$ You can numerically simulate this sum or you can use quadrature methods to approximate the integral. The latter is probably better.