integration - Is there way to integrate this function other than numerically?

I am wondering if there is a way to integrate the function $$\int_{0}^∞ \frac{\tan^{-1}(\pi{x})-\tan^{-1}(x)}{e^{x^2}}$$ without resorting to Simpson's rule/numerical integration. My first thought was to split it into two integrals and use integration by parts on each, but that did not work. Each step integrating by parts just convoluted it more and more.

Depending how you integrate by parts, you either get the chain of convoluted integrals I did, or immediately wind up with the Gauss Error Function $\text{erf}(x)$.

I'm self-taught and still relatively new, so please forgive me if I missed something simple. Anyone know what to do here?

Answer

This is a real monster ... but a CAS gave a result I am sure you will enjoy !

If
$$I_a=\int_0^\infty e^{-x^2} \tan ^{-1}(a x)\,dx$$ then, assuming that $a$ is a real positive number,
$$I_a=\frac{1}{4} \sqrt{\pi } \left(\text{erfi}\left(\frac{1}{a}\right) (\gamma -2 \log
(a))+\pi \right)+$$ $$\frac{\text{HypergeometricPFQ}^{(\{0,0\},\{0,1\},0)}\left(\left\{\frac{1}{2},1\right\},\left\{\frac{3}{2},1\right\},\frac{1}{a^2}\right)-2 \,
_2F_2\left(\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{1}{a^2}\right)}
{2 a} $$where appear hypergeometric functions and derivatives.

Its decimal representation of $(I_\pi-I_1)$ is $0.372711790424249$