$\int\frac{x^n}{1+x^m}dx$

Question

How do I get a closed form of $$\int\frac{x^n}{1+x^m}dx$$

Background: There were many times that, when I put an integral of the form: $$\int_{-\infty}^\infty\frac{x^n}{1+x^m}dx$$in Wolfram Alpha, it gives the indefinite integral too. So I wonder if there is a closed form expression for this indefinite integral.

I know that when $n=m-1$ we easily obtain $\frac{1}{m}\ln(1+x^m)+C$, but other than that I don't know how to obtain the general form. Integration by parts would be too tedious.

https://math.stackexchange.com/questions/110457/closed-form-for-int-0-infty-fracxn1-xmdx Related — Sine of the Time, Feb 07 '23 at 19:42
Does this answer your question? Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ — uniquesolution, Feb 07 '23 at 19:43
That link deals with a definite integral from $0$ to $\infty$. Here OP is asking about the indefinite integral. — Robert Israel, Feb 07 '23 at 19:44
@RobertIsrael I know, in fact I'm not saying it's a dupe, it's just related — Sine of the Time, Feb 07 '23 at 19:45
The wording of the question is potentially misleading. The display equation is a definite integral, but the following statement refers to "this indefinite integral". It would make the question easier to interpret if the indefinite integral were stated explicitly in another display equation. — Travis Willse, Feb 07 '23 at 22:38
For the special case $a, b \in \Bbb N$, you can find an explicit form in terms of elementary functions here: https://math.stackexchange.com/questions/2409312/evaluating-int-fracxb1xadx-for-a-b-in-bbb-n?noredirect=1 — Travis Willse, Feb 08 '23 at 00:56

Robert Israel · Accepted Answer · 2023-02-07T20:03:05.483

4

The antiderivative can be written in terms of the Lerch Phi function.

$$\frac{x^{n +1} \Phi \! \left(-x^{m}, 1, \frac{n +1}{m}\right)}{m}$$

This can also be written as a hypergeometric function:

$$\frac{x^{n +1} {}_{2}F_{1}\left(1,\frac{n +1}{m};\frac{m +n +1}{m};-x^{m}\right)}{n +1} $$

I think that's the best you can do if you want a "closed form" in general. For particular positive integers $m$ and $n$ you can get an "elementary" (but possibly messy) antiderivative by expanding the integrand in partial fractions.

edited Feb 07 '23 at 20:03

answered Feb 07 '23 at 19:47

Robert Israel

448,999

There is also an incomplete beta function conversion in terms of $B_z(a,0)$ – Тyma Gaidash Feb 08 '23 at 00:46

$\int\frac{x^n}{1+x^m}dx$

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