A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Question

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$.

• Is there any algebraic (or at least period) $a\notin\mathbb{Z}^+$ such that $I(a)$ has a closed form representation?

• In particular, does $\displaystyle I\left(\frac12\right)=\int_0^\infty\sqrt{\frac{2^{-x}-3^{-x}\vphantom|}{x}}\ dx\ $ have a closed form representation?