What is the integral of ln(secx + tanx)?

[JamesGregory]

I was taking the integral of the secant function. Twice...
The first one is simple, but what is the integral of
ln(secx + tanx)dx?

I've tried a few things, the first being integration by parts with u = ln(secx + tanx+) and dv = dx
This just cancels in the end to 0 = 0
I also rewrote it as int[ln(1+sinx) - ln(cosx)]dx but that doesn't seem to be any easier.

Any suggestion would be greatly appreciated.

 

[Vid]

It doesn't have a closed form answer in terms of elementary functions.

 

[ice109]

sure it does, unless you consider the polylog erudite:

$$\frac{1}{2} \left(\pi \left(i x+\text{Log}\left[-\text{Sin}\left[\frac{1}{4} (\pi -2 x)\right]\right]-\text{Log}[1+i \text{Cos}[x]-\text{Sin}[x]]-\text{Log}[1-i \text{Cos}[x]+\text{Sin}[x]]+\text{Log}\left[\text{Sin}\left[\frac{1}{4} (\pi +2 x)\right]\right]\right)+2 x \left(2 i \text{ArcTan}\left[e^{i x}\right]+\text{Log}[\text{Sec}[x]+\text{Tan}[x]]\right)+2 i \text{PolyLog}\left[2,i e^{i x}\right]-2 i \text{PolyLog}[2,-i \text{Cos}[x]+\text{Sin}[x]]\right)$$

lol off screen. whatever it's in the body of the post, just click quote or something

 

[Vid]

Elementary functions are finite combinations and compositions of algebraic, logarithmic, and exponential functions. Polylog obviously doesn't fit that description.

 

[ice109]

yea you're right

 

[PolyLog]

@ice109: How do you get that formula? (The one that involves PolyLog)

 

[darkside00]

Here

http://integrals.wolfram.com/index.jsp?expr=ln(secx+tanx)&random=false

 

[PolyLog]

Thank you!