Mathematical reasoning to get closed-forms or nice definite integrals from these outputs of Wolfram Alpha I was thinking about the shape of integrals related with $\zeta(3)$ and Catalan's constant, I am saying those in section 3.1 of this Wikipedia. I was thinking in moments of higher order $x^k$ in the integrand, and since I believe that these integrals will be well known, after I was trying to calculate with Wolfram Alpha these integrals that are different, I am saying this $$\int e^{-i x}\log\left(\sec(x)+\tan(x)\right)dx\tag{1}$$ and

Codes. You can see the different closed-forms that Wolfram Alpha provide us as outputs for these indefinite integrals, involving logarithms and complex exponentials, hypergeometric functions and polylogarithms, and also trigonometric functions:
integrate xe^(-ix) log(sec(x)+tan(x))dx $\tag{2}$
integrate  e^(-i s x) log(sec(x)+tan(x))dx$\tag{3}$
integrate  e^(-i s x) log(1+tan(x))dx$\tag{4}$
integrate  e^(-i s x) log(1+sec(x))dx$\tag{5}$

From the online calculator of Wolfram Alpha, and from my computer with standard computation time, I only obtain as output a definite integral, in example $(1)$, for which I believe that it's easy to prove  $$\Re\left(\int_0^{\pi} e^{-i x}\log\left(\sec(x)+\tan(x)\right)dx\right)=\pi.$$

Question. Imagine that from these kind of integrals $(1)-(5)$ you need to create a nice closed-form, you can take also the real or imaginary part. What is the algebraic/analytic tricks that do you make to explore and exploit your possiblilities from Wolfram Alpha's outputs? I am saying that we need to do some evaluations of the  integration limits, you can take these following your reasoning, but also we need to have knowledges about the functions involved in the outputs. What is the output $(1)-(5)$ that do you choose? What are your manipulations and final  statement? Many thanks.

 A: Partial Answer: My suggested first two steps for your strategy since I can't comment.
Step 1. Look for Mathematical/Trignometric Identities and investigate other work done using those.
In the case of this question it is well worth exploring the available trigonometric identities before looking at substitutions. e.g.
$$\log\left(\sec(x)+\tan(x)\right)= \frac{1}{2} \log\left(\frac{1+\sin(x)}{1-\sin(x)}\right)= \log \tan\left( \frac{\pi}{4}+\frac{x}{2}\right) $$
which is very closely related to
$$\log\left(\csc(x)-\cot(x)\right)=-\log\left(\csc(x)+\cot(x)\right)= \frac{1}{2} \log\left(\frac{1-\cos(x)}{1+\cos(x)}\right)= \log \tan\left(\frac{x}{2}\right) $$
For work on integrating $f(x)\log\tan(x)$ that may help you formulate your strategy in regards to integrating $f(x)\log \tan\left( \frac{\pi}{4}+\frac{x}{2}\right)$ see for example https://arxiv.org/abs/1611.01274
Step 2. Search for Substitutions.
For example the standard integral for the Dirichlet Beta Function can be written in substituted form using $\log \tan (x)$
$$\Gamma(n)\beta(n)=\int_0^\infty \frac{u^{n-1}}{e^u+e^{-u}}\,du= \frac{1}{2}\int_{\pi/4}^{\pi/2} \left( \log \tan x\right)^{n}\,dx$$
This might also help you think about integration limits to be applied.
