Power Series Expansion for Logarithm of Cosine of x

Theorem
for all $x \in \R$ such that $\left|{x}\right| < \dfrac \pi 2$.

Proof
From Power Series Expansion for Tangent Function:

for $\left|{x}\right| < \dfrac \pi 2$.

From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term: