Let $c>0$. In the process of solving a problem related to Gaussian integrals we came across the following integral:
\begin{eqnarray} {\mathcal I}(c) &:=& \int\limits_0^c \frac{t }{\sqrt{2 t^2+1} \left(3 t^2+1\right) \sqrt{4 t^2+1} } \cdot \arctan\left(\sqrt{2 t^2+1}\right) \cdot \arctan\left(\frac{t}{\sqrt{4 t^2+1}}\right) dt \quad (i)\\ \end{eqnarray}
Observation: The left-most term in the integrand has a neat anti-derivative given below:
\begin{equation} \int \frac{t }{\sqrt{2 t^2+1} \left(3 t^2+1\right) \sqrt{4 t^2+1} } dt = \frac{1}{2} \arctan \left(\frac{t^2}{\sqrt{2 t^2+1} \sqrt{4 t^2+1}}\right) \end{equation}
and therefore there is hope that something can be done using integration by parts.
How would you go about solving integral $(i)$ ?