evaluate $\int\ln x\tan x\, \mathrm dx$

Question

How to evaluate $$\int\ln x\tan x\, \mathrm dx \:?$$ I've tried to do integration by parts but after calculations it cancel out the main question.

Answer 1

Please note that the no-closed-form integrals in fact can be further classified as four types:

Type $1$: Can be expressed as infinite series whose its radius of convergence covers on $\mathbb{C}$ or $\mathbb{R}$

Type $2$: Can only be expressed as infinite series whose its radius of convergence covers on finite ranges, to get the results suitable on $\mathbb{C}$ or $\mathbb{R}$ should have cases divisions

Type $3$: Other than Type $1$ and Type $2$ but the software can express them as known special functions, we can only follow the expressions from the software

Type $4$: Other than Type $1$ and Type $2$ and even the software cannot express them as known special functions, if we really want to force to solve them we should only use the formula e.g. this one.

Unfortunately, $\int\ln x\tan x~dx$ is in fact belongs to Type $4$ because of the following reasons:

$1.$ Wolfram fails to solve this integral.

$2.$ The radius of convergence of the power series of $\ln x$ is only $1$ , and $\int x^n\tan x~dx$ still have no-closed-form

$3.$ The radius of convergence of the power series of $\tan x$ is only $\dfrac{\pi}{2}$ and its coefficients have no-closed-form, and we should have infinitely many times of cases divisions in order to get the results suitable on $\mathbb{C}$ or $\mathbb{R}$