Integration with Half space Gaussian I have a problem to solve and I have something that i don't know how to do.

The half-space Gaussian integral is given :

$$\int_{0}^\infty \exp(-ax^2)dx = \frac{1}2 \sqrt{\frac{\pi}{a}}$$

I have to calculate 

$$\int_{0}^\infty \exp \left(-ax^2 - \frac{b}{x^2} \right)dx$$

a and b are real and positive

Is someone have an idea ? :)

Thanks for your answers

Mathieu
Note that in THIS ANSWER, I presented a solution to a more general version of the integral of interest herein.  


Let $I(a,b)$ be the integral given by

for $a&gt;0$ and $b&gt;0$.

$$I(a,b)=\int_0^\infty e^{-\left(ax^2+\frac{b}{x^2}\right)}\,dx \tag 1$$

Enforcing the substitution $x\to \sqrt[4]{b/a}x$ into $(1)$ reveals

$$\begin{align}
I(a,b)&amp;=\sqrt[4]{\frac{b}{a}}\int_0^\infty e^{-\sqrt{ab}\left(x^2+\frac{1}{x^2}\right)}\,dx \tag 2
\end{align}$$

Next, noting that $x^2+\frac{1}{x^2}=\left(x-\frac1x\right)^2+2$, we can write $(2)$ as 

$$\begin{align}
I(a,b)&amp;=\sqrt[4]{\frac{b}{a}}e^{-2\sqrt{ab}}\int_0^\infty e^{-\sqrt{ab}\left(x-\frac{1}{x}\right)^2}\,dx\tag 3
\end{align}$$

Enforcing the substitution $x\to 1/x$ in $(3)$ yields

$$I(a,b)=\sqrt[4]{\frac{b}{a}} e^{-2\sqrt{ab}}\int_0^\infty e^{-\sqrt{ab}\left(x-\frac{1}{x}\right)^2}\,\frac{1}{x^2}\,dx\tag4$$

Adding $(3)$ and $(4)$, we obtain

$$\begin{align}
I(a,b)&amp;=\frac12\sqrt[4]{\frac{b}{a}} e^{-2\sqrt{ab}}\int_{-\infty}^\infty e^{-\sqrt{ab}\left(x-\frac{1}{x}\right)^2}\,d\left(x-\frac{1}{x}\right)\\\\
&amp;=\frac12\sqrt[4]{\frac{b}{a}} e^{-2\sqrt{ab}}\int_{-\infty}^\infty e^{-\sqrt{ab}\,x^2}\,dx\\\\ 
&amp;=\frac1{2\sqrt a} e^{-2\sqrt{ab}}\int_{-\infty}^\infty e^{-x^2}\,dx\\\\
&amp;=\bbox[5px,border:2px solid #C0A000]{\frac{\sqrt{\pi}}{2\sqrt a}e^{-2\sqrt{ab}}}
\end{align}$$
$$\exp \left(-ax^2 - \frac{b}{x^2} \right)$$
$$ = \exp \left( -ax^2\right)*\exp \left( -\frac{b}{x^2} \right)$$
$$=\exp \left( -ax^2\right)*\exp \left( -\frac{b*x^2}{x^4} \right)$$
$$=\exp \left( -ax^2\right)*\exp \left( -bx^2\right)*\exp \left( -\frac{1}{x^4} \right)$$

Now something with partial integration?