<s>[INST] How can I evaluate $\int _0^1\frac{\tan ^{-1}\left(3\sqrt{\frac{a}{4-a}}\right)}{\sqrt{a}\sqrt{4-a}}\:\mathrm{d}a$ I have the integral:
$$\int _0^1\frac{\tan ^{-1}\left(3\sqrt{\frac{a}{4-a}}\right)}{\sqrt{a}\sqrt{4-a}}\:\mathrm{d}a.$$
If I use $u=3\sqrt{\frac{a}{4-a}}$, I get
$$6\int _0^{\sqrt{3}}\:\frac{\tan ^{-1}\left(u\right)}{u^2+9}\mathrm{d}u$$
But I cant get past this. I tried integration by parts but nothing seems to come out from that.
 [/INST] Using $$ \tan^{-1}(x) = \frac{i}{2} \, \ln\left( \frac{1 - i \, x}{1 + i \, x} \right) $$ then
\begin{align}
\int \frac{\tan^{-1}(x)}{x^2 + a^2} \, dx &= \frac{i}{2} \, \left( \int \frac{\ln(1 - i \, x)}{x^2 + a^2} \, dx - \int \frac{\ln(1 + i \, x)}{x^2 + a^2} \, dx \right).
\end{align}
Using
\begin{align}
\int \frac{\ln(1 + i \, x)}{x^2 + a^2} \, dx &= \frac{i}{2 \, a} \, \left[ - \text{Li}_{2}\left(\frac{1+ i \, x}{1 - a}\right) + \text{Li}_{2}\left( \frac{1 + i \, x}{1 + a}\right) + \ln(1 + i \, x) \, \ln\left( \frac{a-1}{a+1} \cdot \frac{a - i \, x}{a + i \, x}\right)  \right] \\
\int \frac{\ln(1 - i \, x)}{x^2 + a^2} \, dx 
&= \frac{i}{2 \, a} \, \left[ - \text{Li}_{2}\left(\frac{1- i \, x}{1 + a}\right) + \text{Li}_{2}\left( \frac{1 - i \, x}{1 - a}\right) + \ln(1 - i \, x) \, \ln\left( \frac{a+1}{a-1} \cdot \frac{a - i \, x}{a + i \, x}\right)  \right]
\end{align}
then the result can be found in closed form with:
\begin{align}
\int \frac{\tan^{-1}(x)}{x^2 + a^2} \, dx &= \frac{-1}{4 \, a} \, \left[ - \text{Li}_{2}\left(\frac{1- i \, x}{1 + a}\right) - \text{Li}_{2}\left(\frac{1 + i \, x}{1 + a}\right) + \text{Li}_{2}\left(\frac{1 - i \, x}{1 - a}\right) + \text{Li}_{2}\left(\frac{1 + i \, x}{1 - a}\right) \\+ \ln(1 + x^2) \, \ln\left(\frac{a+1}{a-1}\right) - 4 \, \tan^{-1}(x) \, \tan^{-1}\left(\frac{x}{a}\right) \right]
\end{align}
The integral with the desired limits is:
$$ \int_{0}^{\sqrt{3}} \frac{\tan^{-1}(x)}{x^2 + 3^2} \, dx = \frac{1}{6} \, \left( \frac{\zeta(2)}{3} - \frac{\ln(2) \, \ln(3)}{2} - \frac{\text{Li}_{2}(-3)}{4} \right). $$
Also:
$$ \int_{0}^{1} \tan^{-1}\left( 3 \, \sqrt{\frac{x}{4-x}} \right) \, \frac{dx}{\sqrt{x \, (4-x)}} = \frac{\zeta(2)}{3} - \frac{\ln(2) \, \ln(3)}{2} - \frac{\text{Li}_{2}(-3)}{4}. $$
</s>