$\int x\tan x$ and the Clausen Function

Question

I have been attempting to evaluate $\int x \tan x \;\mathrm{d} x$.

My first instinct was integration by parts, which produces $-x \ln|\cos x|+\int \ln|\cos x| \;\mathrm{d} x$.

I have read online that the order 2 Clausen function can be used to complete the integration, but the Clausen function applies to the standard logarithm function, and I am unclear on how it extends to the natural logarithm. An explanation of this and a guide as to how I can finish the integral would be greatly appreciated.

Thank you in advance for your time.

Answer 1

The $\log$ function in the order 2 Clausen function is the natural logarithm.

$$\mathrm{Cl}_2(\phi) = -\int\limits_{0}^{\phi} \log_e |2 \sin \frac{x}{2}| \mathrm{d}x$$

Logarithms in calculus expressions are always base $e$, rather than base $10$, unless explicitly stated otherwise.  The subscript is often omitted by lazy authors because this convention is so generally understood.

(Though the whole point of having the $\ln$ function was to avoid such confusion.)