calculus - Integrals with the special functions $Ci(x)$ and $erf(x)$

I'm looking for the solutions of the following two integrals:

$$I_1=\int\limits_0^\infty dx\, e^{-x^2}\text{Ci}(ax)$$
and
$$I_2=\int\limits_0^\infty dx\, e^{-ax}\text{erf}(x)$$
with
$$\text{Ci}(x)=\int\limits_\infty^x\frac{\cos(t)}{t}dt$$
and

$$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int\limits_0^xe^{-t^2}dt$$

Now I'm not 100% sure what is meant by "the solution of the integrals" since these will probably be not-evaluable. But I'm guessing that the question is to reduce the expression to one single special function in stead of the integral of a special function with an elementary function.

Mathematica yields me the answers:
$$I_1=-\frac{\sqrt{\pi}}{4}\Gamma\left(0,\frac{a^2}{4}\right)$$
and
$$I_2=\exp\left(\frac{a^2}{4}\right)\frac{1-\text{erf}(a/2)}{a}$$

A good first step for evaluating these integrals $I_1$ and $I_2$ seemed to fill in the integral representations of these special functions and try to switch te integrals over $x$ and $t$. However this has not (yet) been a success. I also tried to find a differential equation for these integrals, but also this was not so easy to do. Are there any tips/tricks to evaluate these integrals ?

Answer

Hint: Differentiate $I_1(a)$ with regard to a. Similarly, define $~I_2(b)~=~\displaystyle\int_0^\infty e^{-ax}~\text{erf}(bx)~dx,~$ and then differentiate it with regard to b.