<s>[INST] 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
$$\Phi(a,b)=\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}$$
 [/INST] Here are a set of complete solutions, with all cases accounted for.
$$\Phi(a,b)=\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}
\overset{x=\frac{bt}{\sqrt{1-t^2}} }=\int_0^{\frac1{\sqrt{1+b^2}} }\frac{dt}{a^2+(b^2-a^2)t^2}
$$
which leads to the solutions for the three distinctive cases below
\begin{align}
 & \Phi(b<a) = \frac{1}{a\sqrt{a^2-b^2}}\tanh^{-1}\frac{\sqrt{a^2-b^2}}{a\sqrt{1+b^2}}\\
 & \Phi(b=a) = \frac{1}{a^2\sqrt{1+a^2}}\\
 & \Phi(b>a) = \frac{1}{a\sqrt{b^2-a^2}}\tan^{-1}\frac{\sqrt{b^2-a^2}}{a\sqrt{1+b^2}}\\
\end{align}
</s>