<s>[INST] parametric integral relating to hyperbolic function Suppose that $a$ is real number such that $0<a<1$, how can we calculate
$$ I(a)=\int_0^\infty \big(1-\frac{\tanh ax}{\tanh x}\big)dx .$$
As for some speical cases, I can work out $I(1/2)=1$.  Any suggestion to the integral above? Tks.
 [/INST] If you are interested in particular cases:
$$\begin{align}
I\left( \frac15 \right) & = \sqrt{2-\frac{2}{\sqrt{5}}} \, \, \pi \\
I\left( \frac14 \right) & = \frac{\pi}{2} + 1 \\
I\left( \frac13 \right) & = \frac{\sqrt{3}}{3} \, \pi \\
I\left( \frac25 \right) & = \left(\sqrt{2+\frac{2}{\sqrt{5}}} - \frac54 \right) \pi \\
I\left( \frac12 \right) & = 1 \\
I\left( \frac35 \right) & = \frac{10\pi}{3\sqrt3} + \sqrt{1+\frac{2}{\sqrt5}} \, \pi- \sqrt{5+2\sqrt5} \, \pi \\
I\left( \frac23 \right) & = \frac{3\pi}{4} - \frac{\sqrt{3} \, \pi}{3} \\
I\left( \frac34 \right) & = \frac{8\sqrt{3} \,\pi}{27} - \frac{\pi}{2} + \frac13 \\
\end{align}$$
There are also some closed-form for $a \geq 1$:
$$\begin{align}
I(1) & = 0 \\
I(2) & = -\frac{\pi}{4} \\
I(3) & = -\frac{2\sqrt{3} \, \pi}{9} \\
I(4) & = -\frac{\sqrt{2} \, \pi}{4} - \frac{\pi}{8} \\
I(5) & = -\frac{2}{5}\sqrt{1+ \frac{2}{\sqrt5}} \, \pi \\
I(6) & = -\frac{5 \pi}{12} - \frac{\sqrt{3} \, \pi}{9}
\end{align}$$
I've got them by using CAS and some manual simplification.

@spanferkel
$$\begin{align}
I\left(\frac 45 \right) & = -\frac{1}{40} \left(25+48 \sqrt{5-2 \sqrt{5}}-2 \sqrt{2050-352 \sqrt{5}-400 \sqrt{25-11 \sqrt{5}}}\right) \pi \\
I\left(\frac 16 \right) & = \frac{5 \pi }{3 \sqrt{3}}+1
\end{align}$$
The expressions are simplifications of sums with $\arctan$ terms.

@spanferkel
$$\begin{align}
I\left( \frac 18 \right) & = 1+\sqrt{2}\,\pi \\
I\left( \frac 28 \right) & = \frac{\pi}{2} +1\\
I\left( \frac 38 \right) & = -{\frac {16}{27}}\,\sqrt {3}\,\pi+\sqrt {2}\,\pi+\frac 13 \\
I\left( \frac 48 \right) & = 1\\
I\left( \frac 58 \right) & = -\sqrt {2}\,\pi +{\frac {32}{125}}\,\sqrt {25+10\,\sqrt {5}}\,\pi -{\frac {16}{125}}\,\sqrt {25-10\,\sqrt {5}}\,\pi+\frac 15 \\
I\left( \frac 68 \right) & = {\frac {8}{27}}\,\sqrt {3}\,\pi -\frac{\pi}{2} + \frac 13
\end{align}$$
I couldn't find a closed-form of $I(7/8)$. The expressions are coming again from sums in term of $\arctan$.
</s>