What would be a relatively simple method for computing the indefinite integral below?
$\displaystyle \int \frac{dx}{(x^4+1)^2}$
Furthermore, how would one evaluate the following, possibly by detouring the computation of the indefinite integral?
$\displaystyle\int\nolimits_{-\infty}^\infty \frac{dx}{(x^4+1)^2}$
Note: The resolved value of the definite integral (according to wolfram alpha) is $\displaystyle \frac{3\pi}{4\sqrt{2}}$
On
For the second, definite integral, assuming you have computed the first, indefinite integral; you can do a change of variables to obtain a form of the integral where you can use differentiation under the integral sign to compute its value.
More explicitly, let $$f(\alpha) = \int_{-\infty}^\infty \frac{1}{(x^4+\alpha)}\mathrm dx ,$$ then the value you want to compute is $$\int_{-\infty}^\infty \frac{1}{(x^4+1)^2}\mathrm dx = - \left.\frac{\mathrm d f}{\mathrm d\alpha}\right|_{\alpha = 1}.$$
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The definite integral can be done using residues. Note that the zeros of $(z^4+1)^2$ in the upper half plane are at $\frac{\pm 1 + i}{\sqrt{2}}$, both with multiplicity 2. The residues are $\frac{3 \sqrt{2}}{32}(1-i)$ at $\frac{-1+i}{\sqrt{2}}$ and $\frac{3 \sqrt{2}}{32}(-1-i)$ at $\frac{1+i}{\sqrt{2}}$ (note that the residue of $\frac{1}{((x-r) f(x))^2}$ at $x=r$, where $f$ is analytic at $r$ with $f(r) \ne 0$, is $\frac{-2 f'(r)}{f(r)^3}$). So the integral is $2 \pi i \frac{3 \sqrt{2}}{32} (1-i -1 - i) = \frac{3 \pi \sqrt{2}}{8}$.
Well, the problem is tagged "partial fractions," that seems to me to be the way to go. Of course, first you need to factor $x^4+1$ into irreducible quadratics; that can be done by $$x^4+1=x^4+2x^2+1-2x^2=(x^2+1)^2-(\sqrt2x)^2$$
It will be messy - do you have any reason to think there is a simpler method?
Edit: for the definite integral, if you've done complex variables, you'll know about contour integration, which ought to handle this example without difficulty. If you haven't done complex variables, you have something truly wonderful to look forward to.