Does there exist a (preferably colsed-form, i.e. not an infinite series) solution to:
$$\int_{0}^{t} e^{f(x)} dx$$
where $f(x)$ is a polynomial of degree $3$, let's say $f(x) = a + bx + cx^2 + dx^3$
I've tried expanding the integrand using a McLauren series, thinking that $f^{(n)}(x) = 0$ for $n>3$ and $f^{(n)}(0)$ will be a function of at most one of $\{a,b,c,d\}$, the polynomial coeficients, could perhaps simplify things. However, it seems the recursive nature of derivatives of exponentials of functions just keeps spitting out functions of the lower order derivatives of $f(x)$. This lead me to briefly try finding a recursive formula which might be evaulable, or recognized as a Taylor series of some function, to no avail.
