Pi as Sum of Alternating Sequence of Products of 3 Consecutive Reciprocals/Lemma

Theorem

 * $\displaystyle \int \int \int \dfrac x {x^2 + 1} \rd x \rd x \rd x = x \, \map \arctan x + \dfrac {\paren {x^2 - 1} \, \map \ln {x^2 + 1} - 3 x^2} 4$

with all integration constants at $0$.

Proof
First Integral:

The integration constant is not added due to the series never having a constant during its integration.

Second Integral:

Third Integral: