Closed form of $\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}$ Is it possible to get a closed form of the following integral
$$\Phi(a,b)=\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}$$

For real $a,b\neq 0$ Mathematica gives
$$
  \Phi(a,b) = \frac{1}{a\sqrt{b^2-a^2}}\cdot\arctan\left(\frac{\sqrt{b^2-a^2}}{a\sqrt{1+b^2}}\right)
$$
