Elliptic integral of a quartic

Question

I am trying to compute the following integral: \begin{equation} I(a) = \int_{-\infty}^{+\infty} \left[ 1-x^2 + \sqrt{(1-x^2)^2 + a} \right] dx \end{equation} where $a >0$.

I am pretty confident this integral is well defined, as it basically looks like a bell-shaped curve that behaves like $\frac{1}{x^2}$ as $x$ gets large.

More precisely, I am trying to get the first non-constant term of the Taylor expansion of $I(a)$ as $a \to 0$.

Answer 1

With CAS help:

$$\int_{-\infty }^{\infty } \left(1-x^2+\sqrt{\left(1-x^2\right)^2+a}\right) \, dx=\\\frac{4}{3} \sqrt[4]{1+a} \left(2 E\left(\frac{1}{2} \left(1+\frac{1}{\sqrt{1+a}}\right)\right)+\left(-1+\sqrt{1+a}\right) K\left(\frac{1}{2} \left(1+\frac{1}{\sqrt{1+a}}\right)\right)\right)$$

for $a > 0 $.

where: $E$ and $K$ Elliptic Integral of the Second Kind and Complete Elliptic Integral of the First Kind

Mathematica code:

  HoldForm[Integrate[1 - x^2 + Sqrt[(1 - x^2)^2 + a], {x, -Infinity, Infinity}] == 
  4/3 (1 + a)^(1/4) (2 EllipticE[1/2 (1 + 1/Sqrt[1 + a])] + 
  (-1 + Sqrt[1 + a]) EllipticK[1/2 (1 + 1/Sqrt[1 + a])])] // TeXForm

  Series[4/3 (1 + a)^(1/4) (2 EllipticE[
  1/2 (1 + 1/Sqrt[1 + a])] + (-1 + Sqrt[1 + a]) EllipticK[
  1/2 (1 + 1/Sqrt[1 + a])]), {a, 0, 2}, 
  Assumptions -> a > 0] // Normal

Expanding with series at $a=0$

$I(a)=\frac{8}{3}+\frac{1}{2} a (1+4 \ln (2)+\log (4)-\ln (a))+\frac{1}{128} a^2 (17-24 \ln (2)-6 \ln (4)+6 \ln (a))+\text{...}$