Can the integral of tan(logx) be simplified for real values of x?

[hsostwal]

What is integral of tan(logx)? I couldn't find it on internet.

 

[CompuChip]

It doesn't seem to be expressible in elementary functions, at best you will get Euler Beta functions.

 

[gnurf]

Did you try http://www70.wolframalpha.com/input/?i=integral+of+tan(log(x))"?

 

[qntty]

Were you just wondering, or do you need to find it for an assignment? Because you should be aware that if you just come up with an integral with elementary functions, there's a good chance that it won't be expressible in terms of elementary functions.

 

[g_edgar]

What is integral of tan(logx)? I couldn't find it on internet.

Maple does it in terms of the Lerch $$\Phi$$ function.

 

[Gregg]

Mathematica:

$-i \left(\left(\frac{1}{5}-\frac{2 i}{5}\right) x^{1+2 i} \text{Hypergeometric2F1}\left[1-\frac{i}{2},1,2-\frac{i}{2},-x^{2 i}\right]-x \text{Hypergeometric2F1}\left[-\frac{i}{2},1,1-\frac{i}{2},-x^{2 i}\right]\right)$

 

[clustro]

A wonderful case of the mathematics required to understand the problem far-exceeding that required to solve it O_O.

 

[CompuChip]

Mathematica:

$-i \left(\left(\frac{1}{5}-\frac{2 i}{5}\right) x^{1+2 i} \text{Hypergeometric2F1}\left[1-\frac{i}{2},1,2-\frac{i}{2},-x^{2 i}\right]-x \text{Hypergeometric2F1}\left[-\frac{i}{2},1,1-\frac{i}{2},-x^{2 i}\right]\right)$

You might want to use FullSimplify (possibly with the additional assumption that x is real) and arrive at something with Euler B's, as I said.

 

[Gregg]

You might want to use FullSimplify (possibly with the additional assumption that x is real) and arrive at something with Euler B's, as I said.

$\text{FullSimplify}[\text{Assuming}[x\in \text{Reals},\int \text{Tan}[\text{Log}[x]] \, dx]]$

$i x \text{Hypergeometric2F1}\left[-\frac{i}{2},1,1-\frac{i}{2},-x^{2 i}\right]-\left(\frac{2}{5}+\frac{i}{5}\right) x^{1+2 i} \text{Hypergeometric2F1}\left[1,1-\frac{i}{2},2-\frac{i}{2},-x^{2 i}\right]$

I'm just copying what it says. I have no idea about all the non elementary functions that pop out or how to see them without resorting to mathematica. e.g. integrating gaussian distribution thing. A source for this information would be good.

 

[CompuChip]

You might want to simplify with the assumption that x is real:

$$\text{Assuming}[x \in \text{Reals}, \text{FullSimplify}[\int \tan[\log[x]], x]]$$

gives
$$\frac{1}{2} \left(-x^{2 i}\right)^{\frac{i}{2}} x \left(B_{-x^{2 i}}\left(-\frac{i}{2},0\right)+B_{-x^{2 i}}\left(1-\frac{i}{2},0\right)\right)$$

which for evaluation purposes doesn't help you, it's just prettier (or less ugly, if you like) to look at.