I want to symbolically write (in the form of a series), the integral of:
$$ \int_{a}^{b} e^{x^2}(\textrm{erf}(x) - \textrm{erf(a)})\;\textrm{d}x, \text{where }\{x, a, b\} \subset \mathbb{R} $$
The $\textrm{erf}$ function is:
$$ \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\;\textrm{d}t$$
The first idea I had is to write the integrand in terms of its power series. However, the product of two infinite series is very messy, and unlikely to be easy to write term-by-term, and thus we can't integrate some nice series term-by-term.
Let us say that I could write the integrand prettily, then there is still the issue of which point I should write the Taylor series expansion of the integrand about, given that it needs to be an accurate representation of the function about all points in the range of integration. Perhaps the infinite number of terms suffices here?
Well, that about exhausts my meagre symbolic integration capacities. Note that I do not want to compute this numerically. Wolfram Alpha runs out of computation time when given the input: integrate exp(x^2)*(erf(x) - erf(a)) dx from a to b
However, when Wolfram Alpha is given the indefinite calculation: integrate exp(x^2)*(erf(x) - erf(a)) dx, it provides us with the result that uses hypergeometric functions, and the imaginary $\textrm{erf}$ function. It is easy to use the indefinite integral to compute the definite integral (Fundamental Theorem of Calculus), but I'd like to ask: if I didn't have Wolfram Alpha, and only pen and paper, how would I go about symbolically computing the indefinite integral?