# Integral of tan(logx)

1. Jun 21, 2009

### hsostwal

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

2. Jun 21, 2009

### CompuChip

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

3. Jun 21, 2009

### gnurf

Last edited by a moderator: Apr 24, 2017
4. Jun 21, 2009

### 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.

5. Jun 25, 2009

### g_edgar

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

6. Jun 28, 2009

### Gregg

Mathematica:

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

7. Jun 28, 2009

### clustro

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

8. Jun 29, 2009

### CompuChip

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.

9. Jun 29, 2009

### Gregg

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

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

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.

10. Jun 30, 2009

### 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.