Evaluate this definite integral:
n
∫ x^{2}*(1+x^{4})^{-1} dx
1/n
where n is a real number greater than 1.

Integ {1/n, n} x^2/(x^4+1) dx

Let's start with the substitution y=1/x. then dy=-1/(x^2) n->1/n and 1/n->n. Then after some simplification we have

Integ {1/n, n} 1/(y^4+1) dy

Since this is a definite integral, y is just a dummy variable and can be changed back to x.

This result is very useful when we combine the two integrals by taking their average to make

1/2 * Integ {1/n, n} (x^2+1)/(x^4+1) dx

(x^2+1)/(x^4+1) has a rather simple partial fraction decomposition of

(1/2)/(x^2+sqrt(2)x+1) + (1/2)/(x^2-sqrt(2)x+1)

Then I'll complete the square and substitute this into the integral

1/4 * Integ {1/n, n} 1/[(x+1/sqrt(2))^2+(1/sqrt(2))^2] + 1/[(x-1/sqrt(2))^2+(1/sqrt(2))^2] dx

Now both terms of the integral can be integrated using arctan. Doing so and simplifying a bit yields

1/(2*sqrt(2)) * (arctan[sqrt(2)*x+1] + arctan[sqrt(2)*x-1]) |{1/n, n}

At this point I want to apply the arctan addition formula to simplify stuff, but it doesn't quite work cleanly when the arguments are outside of the interval (-1,1)