integrals with error function Can anyone help me to compute these integrals?
\begin{equation}
\int_0^t\frac{1}{x}\exp\left(-\frac{a^2}{x}\right)
\operatorname{erf}\left(\frac{b}{\sqrt{x}}\right)\,dx
\end{equation}
here $\operatorname{erf}(\cdot)$  is error function.
I have already tried to take this integral by a lot of means, but I have not succeeded... 
There is one more (it is consequence of the first one):
\begin{equation}
\int_0^b\frac{1}{\sqrt{a^2 + x^2}}
\operatorname{erfc}\left(\sqrt{\frac{a^2 + x^2}{t}}\right)\,dx
\end{equation}
Here $\operatorname{erfc}(\cdot)$ is complementary error function.
Of course, I understand, that it is impossible to take them in elementary functions... But even with special functions I can't understand how to do it...
Thanks in advance!
 A: \begin{eqnarray}
\int\limits_0^t \frac{1}{x} \exp\left( -\frac{a^2}{x} \right) Erf\left( \frac{b}{\sqrt{x}}\right) dx=\\
4 \int\limits_0^b \Phi^>(\sqrt{\frac{2}{t}} \sqrt{a^2+x^2}) \frac{1}{\sqrt{a^2+x^2}} dx=\\
4 \int\limits_0^{arcsinh(b/a)} \Phi^>(\sqrt{\frac{2}{t}a \cosh(x)}) dx=\\
4 arcsinh(\frac{b}{a}) - 4 \int\limits_0^{arcsinh(\frac{b}{a})} \Phi^<(\sqrt{\frac{2}{t}a \cosh(x)}) dx=\\
2 arcsinh(\frac{b}{a}) - 4\left(\frac{1}{\sqrt{2 \pi}} \sum\limits_{n=1}^\infty (-\frac{1}{2})^{n-1} \frac{1}{(n-1)! (2n-1)} (\frac{2 a}{t})^{n-1/2} \int\limits_0^{arcsinh(\frac{b}{a})} [\cosh(x)]^{2n-1} dx\right)
\end{eqnarray}
In the first line we differentiated the integral with respect to $b$ then carried out the resulting integral by substituting for $1/\sqrt{x}$ and then integrated back again. In the second line we substituted $x\leftarrow a \sinh(x)$ and in the final line we expanded the CDF in a Taylor series .
In order to finish the calculation it amounts to use the following result:
\begin{equation}
\int [\cosh(x)]^n dx = \frac{1}{2^{n-1}} \sum\limits_{p=0}^{2n-1} \binom{2n-1}{p} \frac{1}{2n-1-2 p} \sinh((2n-1-2 p) x)
\end{equation}
