Evaluate $\int \frac {x^2}{\sqrt{\arctan x}} dx$

Question

Is there any closed form expression of $$\int \dfrac {x^2}{\sqrt{\arctan x}} dx?$$

Answer 1

Is there any closed form expression

No, there is no closed form expression in terms of standard mathematical functions (including the known special functions).

I hightly doubt there is any nice closed form definite integral for this function. We can transform the integral the following way:

$$\int \frac{x^2}{\sqrt{\arctan x}} dx=\int \frac{1}{\sqrt{u}} \tan^2 u (1+\tan^2 u)du=2\int \tan^2 v^2 (1+\tan^2 v^2)dv$$

Now look at the first (more simple) integral:

$$\int \tan^2 v^2 dv$$

The closest thing we have are Fresnel integrals:

$$\int_0^x \sin t^2 dt, \qquad \int_0^x \cos t^2 dt $$

But these integrals converge for $x \to \infty$ and this is the only limt at which they have nice closed forms.

Our integral doesn't converge at infinity and it's unlikely to be represented by Fresnel integrals or any known special functions.