All Questions
Tagged with polylogarithm indefinite-integrals
12
questions
2
votes
1
answer
83
views
How to solve the indefinite integral $\int x \cot x\,\mathrm dx$
We can easily solve it with the bounds $0$ to $\frac{\pi}{2}$, but how to solve the indefinite integral?
Wolfram Alpha gives the following solution:
$$\int x \cot(x)\,\mathrm dx = x \log\left(1 - e^{2 ...
- 133
3
votes
1
answer
242
views
How to solve $\int\frac{x\arctan x}{x^4+1}dx$ in a practical way
I need to evaluate the following indefinite integral for some other definite integral
$$\int\frac{x\arctan x}{x^4+1}dx$$
I found that
$$\int_o^\infty\arctan{(e^{-x})}\arctan{(e^{-2x})}dx=\frac{\pi G}{...
- 2,370
1
vote
0
answers
117
views
Closed-form for $\int_0^{a^2} \mathrm{Ei} (-s) \frac{1 - e^s}{s} ds$
In my partial answer to this question: Integral involving polylogarithm and an exponential, I arrive at the integral
$$ \int_0^{a^2} \mathrm{Ei} (-s) \frac{1 - e^s}{s} ds , ~~~~ (\ast) $$
where $a \in ...
- 753
3
votes
0
answers
290
views
Evaluate $\int_{1}^{\infty}\frac{\operatorname{Li}_3(-x)\ln(x-1)}{1+x^2}\text{d}x$
Using $$
\operatorname{Li}_3(-x)
=-\frac{x}{2}\int_{0}^{1}\frac{\ln^2t}{1+tx}
\text{d}t
$$
It might be
$$
-\frac{1}{2}\int_{0}^{1}\ln^2t
\int_{1}^{\infty}\frac{x\ln(x-1)}{(1+tx)(1+x^2)}\text{d}x\text{...
- 5,362
5
votes
1
answer
180
views
Is a closed form possible for $\int\frac{\text{Li}_2(x)^2}{x}dx$?
Can $\,\displaystyle\int\frac{\text{Li}_2(x)^2}{x}dx\,$ be calculated
by a sum/term of polylogarithm functions and the natural logarithm and polynomials (“closed form”) ?
For the special case $\,\...
- 11.6k
4
votes
2
answers
167
views
I need help computing $\int {\ln x\over 2-x}\, dx$
While integrating $\ln(\sec x)$, at one point I managed to break the integral into two. But I wasn't able to integrate one of those parts.
The integral I am having a difficulty with is:
$$\int {\ln ...
- 3,865
3
votes
1
answer
226
views
The indefinite integral $\int\frac{\operatorname{Li}_2(x)}{1+\sqrt{x}}\,dx$: what is the strategy to get such indefinite integral
Here there is an integral that I've found playing with Wolfram Alpha online calculator (thus to me is a curiosity that it has indefinite integral) $$\int\frac{\operatorname{Li}_2(x)}{1+\sqrt{x}}\,dx,\...
user243301
2
votes
1
answer
182
views
About the integral $\int\arctan\left(\frac{1}{\sinh^2 x}\right)dx$, some idea or feedback
While I was playing with Wolfram Alpha calculator I wondered if it is known a closed-form for $$\int_0^\infty\arctan\left(\frac{1}{\sinh^2 x}\right)dx.\tag{1}$$
Wolfram Alpha provide me the ...
user243301
1
vote
1
answer
709
views
Integration of a polylogarithm: Is this function known?
I would like to integrate a polylogarithm of a given order
$$\int dx \mbox{Li}_{n-1}(x)$$
suppose that the order is $n\le 0$ and $x\in(-\infty,0]$, so the function is bounded. I know that it can be ...
- 1,742
6
votes
1
answer
178
views
Evaluating $\int \arccos\bigl(\frac{\cos (x)}{r}\bigr)\sin^2(x){\mathrm dx}$
Following from the previous question
Evaluating $\int \arccos\bigl(\frac{\cos(x)}{r}\bigr) \, \mathrm{d}x$
I now need the extra $\sin^2x$ as in the title. Of course one power of $\sin(x)$ is easy, ...
user200629
8
votes
2
answers
391
views
Closed form for $\int z^n\ln{(z)}\ln{(1-z)}\,\mathrm{d}z$?
Problem. Find an anti-derivative for the following indefinite integral, where $n$ is a non-negative integer:
$$\int z^n\ln{\left(z\right)}\ln{\left(1-z\right)}\,\mathrm{d}z=~???$$
My attempt:
...
- 30.7k
4
votes
3
answers
270
views
Tough quadrilogarithm integral
Solve the follolwing definite integral
$$\int \frac{\operatorname{Li}_4(z)}{1-z}\, dz$$
It is easy for lower powers!
- 14.4k