Integral of the form $\int_0^\infty \exp{\left(-\frac{(a+bx)^2}{x}\right)}\frac{dx}{\sqrt{x^3}}$

I am currently working on the following improper integral:

Let $$\sigma>0$$ and $$H>1$$. I would like to show that $$\int_0^{\infty} \frac{\frac{\ln H}{\sigma}\exp \left( \frac{- \left( \large \frac{\ln H}{\sigma}+ \frac{t \sigma}{2} \right)^2}{2t} \right)}{\sqrt{2 \pi t^3}} \,dt = \frac{1}{H}.$$

This is the result indicated in my book and also on Wolfram Alpha, but I do not know the substitution required to evaluate this. Any ideas?

• I am not sure that an antiderivative could be found (even with special functions). Effectively $$\int_0^\infty \frac{e^{-\frac{(a+b t)^2}{t}}}{t^{3/2}}\,dt=\frac{\sqrt{\pi } e^{-4 a b}}{a}$$ – Claude Leibovici Sep 29 at 6:37
• @ClaudeLeibovici Thanks. May you please explain how to get this formula? Contour integration? I still cannot derive this.... – Richard Sep 29 at 6:41
• No idea and I am sorry for that. This is a result given by a CAS. – Claude Leibovici Sep 29 at 6:44
• I would say that the original integral is quite scary at first sight, but if you set $a=\frac{\ln H}{\sqrt 2 \sigma}$ and $b=\frac{\sigma}{2\sqrt 2}$, as Claude did, then things become much friendlier. – Nyssa Sep 29 at 8:57

$$I(a,b)=\int_0^\infty \exp{\left(-\frac{(a+bx)^2}{x}\right)}\frac{dx}{\sqrt{x^3}}\overset{x\to x^2}=2\int_0^\infty \exp{\left(-\frac{(a+bx^2)^2}{x^2}\right)}\frac{dx}{x^2}$$ $$\overset{\large \frac{1}{x^2}\to x}=\color{blue}{2\int_0^\infty \exp{\left(-{(ax+b/x)^2}\right)}dx}\overset{\large x\to \frac{b}{ax}}=\color{red}{\frac{2b}{a}\int_0^\infty \exp\left(-(ax+b/x)^2\right)\frac{dx}{x^2}}$$ $$\color{purple}{2I(a,b)}=\color{purple}{2\int_0^\infty \exp\left(-(ax+b/x)^2\right)}\left(\color{blue}{1}+\color{red}{\frac{b}{ax^2}}\right)dx$$

It would have been perfect if we had $$(ax-b/x)$$ instead of $$(ax+b/x)$$ since it's derivative would be found in $$\left(1+b/(ax^2)\right)$$, but we can adjust things:

$$(ax+b/x)^2=a^2 x^2+b^2/x^2+2ab$$ $$(ax-b/x)^2=a^2x^2+b^2/x^2-2ab$$ $$\Rightarrow (ax+b/x)^2=(ax-b/x)^2+4ab$$

$$\Rightarrow I(a,b)=\frac{e^{-4ab}}{a}\int_0^\infty \exp\left(-(ax-b/x)^2\right)\left(a+\frac{b}{x^2}\right)dx$$ $$\overset{ax-b/x=t}=\frac{e^{-4ab}}{a}\int_{-\infty}^\infty e^{-t^2}dt=\frac{\sqrt{\pi}e^{-4ab}}{a}$$

• This is really nice ! Thanks for posting such a beautiful anser and $+1$ for sure. – Claude Leibovici Sep 29 at 8:57
• Thank you Claude! Regarding the comments from bellow the question, wolfram can produce a closed form in terms of the error function for $\int \exp{\left(-{(ax+b/x)^2}\right)}dx$ so the original integral must have an antiderivative too. Not sure why it didn't spit one out initially. – Nyssa Sep 29 at 9:02
• I got it thanks to your work. Cheers :-) – Claude Leibovici Sep 29 at 9:24
• At the price of a bunch of $\sqrt{a^2}$ and $\sqrt{b^2}$ we can cover all situations but $(a,b>0)$ give the simplest formula. – Claude Leibovici Sep 29 at 12:49

After カカロット's elegant solution, assuming $$a>0$$ and $$b>0$$, we can find $$\int \frac{e^{-\frac{(a+b x)^2}{x}}}{x^{3/2}} \, dx=\frac{\sqrt{\pi } e^{-4 a b}}{2 a} \left(\text{erfc}\left(\frac{a-b x}{\sqrt{x}}\right)+e^{4 a b} \text{erfc}\left(\frac{a+b x}{\sqrt{x}}\right)-2 \right)$$