An iterative quotient integration

[Wu Xiaobin]

I have got this integration:
$\int_{-1}^{1}\frac{x^n}{(1+w^2-P^2-2wx+p^2x^2)^2}dx$
And at the same time, I can provide several results when n is small.
For example:
when $n=1$, the result is
$\frac{2w}{(1-P^2)(1-w^2)}$
when $n=2$, the result is
$-\frac{2(1-P^2-w^2)}{P^2(1-P^2)(1-w^2)}+\frac{\Delta}{P^3}$
when $n=3$, the result is
$\frac{2w(3w^2-2w^2P^2+P^4+2P^2-3)}{P^4(1-P^2)(1-w^2)}+\frac{3w\Delta}{P^5}$
where
$\Delta=\sinh^{-1}{\frac{P^2+w}{((1-P^2)(P^2-w^2))^{1/2}}}+\sinh^{-1}{\frac{P^2-w}{((1-P^2)(P^2-w^2))^{1/2}}}$
I have tried it in Mathematica and Maple, However the software can't figure it out.
Does anyone feel familiar with this kind of integration and give me some suggestion?

Look forward to your kind reply!
Sincerely Yours
Jacky Wu

 

[jvicens]

Dear Jacky Wu,

Thank you for sharing your integration problem with us. It seems like you have already made some progress by obtaining several results for different values of n. This can be a good starting point for solving the integration.

One approach you can try is to use the method of partial fractions. This involves breaking down the integrand into smaller fractions that are easier to integrate. You can then use the results you have obtained for n=1,2,3 to determine the coefficients of these smaller fractions.

Another approach is to use integration by parts. This method involves splitting the integrand into two parts and using the formula \int u dv = uv - \int v du. You can choose the parts in such a way that the resulting integral is easier to solve.

If these methods do not work, you can also try using trigonometric substitutions or other advanced techniques such as contour integration or residues.

I would also recommend reaching out to a colleague or a professor who specializes in integration or to a math forum where you can get more specific and detailed guidance on solving your integration. Good luck!