Does the following integral admit a closed-form expression?
$$\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx \;\; , \;\; 0 \leq \alpha \leq 1.$$
Mathematica and Maple couldn't derive solutions in terms of elementary functions. The numerical value with $\alpha = 1$ is approx. 0.361.