How would I integrate $\int_{-\infty}^{\infty} e^{-(x+\tan(x))^2} \mathrm{d}x$?

Question

Around a couple months ago, I found an interesting integral which I haven't been able to solve yet - it goes as follows:

$$\int_{-\infty}^{\infty} e^{-(x+\tan(x))^2} \mathrm{d}x = \sqrt\pi$$

I've attempted different techniques such as Feynman's technique, Laplace Transform, substitutions, integration by parts, and many more - yet none could crack it.

Further, verifying the result via Wolfram Alpha was unsuccessful.

Answer 1

We want to find $$\int_{-\infty}^{\infty} e^{-(x+\tan(x))^2} \mathrm{d}x$$

Now, consider the theorem stated below:

Let $\phi(z)$ be any meromorphic function over $\mathbb{C}$ which

1. preserve the extended real line $\mathbb{R}^* = \mathbb{R} \cup \{ \infty \}$ in the sense: $$\begin{cases}\phi(\mathbb{R}) \subset \mathbb{R}^*\\ \phi^{-1}(\mathbb{R}) \subset \mathbb{R}\end{cases} \quad\implies\quad P \stackrel{def}{=} \phi^{-1}(\infty) = \big\{\, p \in \mathbb{C} : p \text{ poles of }\phi(z)\,\big\} \subset \mathbb{R} $$
2. Split $\mathbb{R} \setminus P$ as a countable union of its connected components $\,\bigcup\limits_{n} ( a_n, b_n )\,$. Each connected component is an open interval $(a_n,b_n)$ and on such an interval, $\phi(z)$ increases from $-\infty$ at $a_n^{+} $ to $\infty$ at $b_n^{-}$.
3. There exists an ascending chain of Jordan domains $D_1, D_2, \ldots$ that cover $\mathbb{C}$, $$\{ 0 \} \subset D_1 \subset D_2 \subset \cdots \quad\text{ with }\quad \bigcup_{k=1}^\infty D_k = \mathbb{C} $$ whose boundaries $\partial D_k$ are "well behaved", "diverge" to infinity and $| z - \phi(z)|$ is bounded on the boundaries. More precisely, let $$ \begin{cases} R_k &\stackrel{def}{=}& \inf \big\{\, |z| : z \in \partial D_k \,\big\}\\ L_k &\stackrel{def}{=}& \int_{\partial D_k} |dz| < \infty\\ M_k &\stackrel{def}{=}& \sup \big\{\, |z - \phi(z)| : z \in \partial D_k \,\big\} \end{cases} \quad\text{ and }\quad \begin{cases} \lim\limits_{k\to\infty} R_k = \infty\\ \lim\limits_{k\to\infty} \frac{L_k}{R_k^2} = 0\\ M = \sup_k M_k < \infty \end{cases} $$

Given such a meromorphic function $\phi(z)$ and any Lebesgue integrable function $f(x)$ on $\mathbb{R}$, we have following identity: $$ \int_{-\infty}^\infty f(\phi(x)) dx = \int_{-\infty}^\infty f(x) dx $$

A proof of this theorem can be found here.

Taking $\phi(x) = x+\tan(x)$ and $f(x) = e^{-x^2}$, we can see that $$\int_{-\infty}^{\infty} e^{-(x+\tan(x))^2} \mathrm{d}x = \int_{-\infty}^{\infty} e^{-x^2} \mathrm{d}x = \sqrt\pi$$

The last integral is a standard Gaussian integral.