I would like to evaluate these two integrals
$$ I_1= \int\limits_0^{+\infty} x \tanh^{-1}\left(\frac{k x}{a}\right) \exp\left(-bx^2\right) dx$$ and $$ I_2=\displaystyle \int\limits_0^{+\infty} x^3 \tanh^{-1}\left(\frac{k x}{a}\right) \exp\left(-bx^2\right) dx$$
$k$ and $b$ are reals.
By Mathematica I get
$$ I_1=\frac{\pi}{4 b} \left[\text{erfi}\left(\frac{a \sqrt{b}}{k}\right)+i\right]e^{-\frac{a^2 b}{k^2}} \,\,\, \text{if} \,\,\,\Im(a)<0 $$
$$I_2=\frac{\sqrt{\pi}}{4 b^2 k^2} \left[-a \sqrt{b} k+\sqrt{\pi } e^{-\frac{a^2 b}{k^2}} \left(a^2 b+k^2\right) \left(\text{erfi}\left(\frac{a \sqrt{b}}{k}\right)+i\right)\right] \,\,\,\text{ if } \,\,\,\Im(a)<0$$
where $\text{erfi}(x)$ denotes the imaginary error function.
How to find these results please?