How to find exact value of integral $\int_{0}^{\infty} \frac{1}{\left(x^{4}-x^{2}+1\right)^{n}}dx$? When I first encountered the integral $\displaystyle \int_{0}^{\infty} \frac{1}{x^{4}-x^{2}+1} d x$, I am very reluctant to solve it by partial fractions and search for any easier methods.  Then I learnt a very useful trick to evaluate the integral.
Noting that
$$I(1):= \int_{0}^{\infty} \frac{d x}{x^{4}-x^{2}+1} \stackrel{x \mapsto \frac{1}{x}}{=} \int_{0}^{\infty} \frac{x^{2}}{x^{4}-x^{2}+1}
$$
Combining them yields
\begin{aligned}
I(1)&=\frac{1}{2} \int_{0}^{\infty} \frac{x^{2}+1}{x^{4}-x^{2}+1} d x\\&= \frac{1}{2}\int_{0}^{\infty} \frac{1+\frac{1}{x^{2}}}{x^{2}+\frac{1}{x^{2}}-1} d x \\
&= \frac{1}{2}\int_{0}^{\infty} \frac{d\left(x-\frac{1}{x}\right)}{\left(x-\frac{1}{x}\right)^{2}+1} \\
&= \frac{1}{2}\tan ^{-1}\left(x-\frac{1}{x}\right)_{0}^{\infty} \\
&= \frac{\pi}{2}
\end{aligned}
Later, I started to investigate the integrands with higher powers.
Similarly,
$$
I(2):= \int_{0}^{\infty} \frac{d x}{\left(x^{4}-x^{2}+1\right)^{2}} \stackrel{x \mapsto \frac{1}{x}}{=}\int_{0}^{\infty} \frac{x^{6}}{\left(x^{4}-x^{2}+1\right)^{2}} d x
$$
By division, we decomposed $x^6$ and obtain $$
\frac{x^{6}}{\left(x^{4}-x^{2}+1\right)^{2}}=\frac{x^{2}+1}{x^{4}-x^{2}+1}-\frac{1}{\left(x^{4}-x^{2}+1\right)^{2}}
$$$$
I(2)=\int_{0}^{\infty} \frac{x^{2}+1}{x^{4}-x^{2}+1} d x-\int_{0}^{\infty} \frac{1}{\left(x^{4}-x^{2}+1\right)^{2}}dx
$$
We can now conclude that $$I(2)=I(1)=\frac{\pi}{2} $$
My Question:
How about the integral $$\displaystyle I_{n}=\int_{0}^{\infty} \frac{d x}{\left(x^{4}-x^{2}+1\right)^{n}}$$ for any integer $n\geq 3$?

Too long for a comment
You could be interested by
$$\displaystyle I_{n}=\int_{0}^{t} \frac{d x}{\left(x^{4}-x^{2}+1\right)^{n}}=t \, F_1\left(\frac{1}{2};n,n;\frac{3}{2};e^{+\frac{i \pi }{3}} t^2,e^{-\frac{i \pi
   }{3}} t^2\right)$$ where appears the Appell hypergeometric function of two variables and the roots of the quadratic in $x^2$.
I did not find any interesting reduction formula or any asymptotics for $t \to \infty$.
Nevertheless,
$$\displaystyle J_{n}=\int_{0}^{\infty} \frac{d x}{\left(x^{4}-x^{2}+1\right)^{n}}=\pi \,\frac {a_n}{2^{b_n}}$$ where the $a_n$ correspond to the sequence
$$\{1,1,9,21,25,483,2359,2907,115533,288915,363363,3674619,37326471,\cdots\}$$ and the $b_n$ the sequence
$$\{1,1,4,5,5,9,11,11,16,17,17,20,23,\cdots\}$$ None of these has been found in $OEIS$.
Edit
For
$$I_{m,n}=\int_0^\infty\frac{dx}{(x^m+1)^n}=\frac{\Gamma \left(1+\frac{1}{m}\right) \Gamma \left(n-\frac{1}{m}\right)}{\Gamma(n)} \qquad \text{if} \qquad \Re(m n)>1\land \Re(m)>0$$
gives the simple
$$I_{m,n+1}=\left(1-\frac{1}{m n}\right)I_{m,n}$$
