How to integrate $\frac{1}{1+x^4}$

Question

Evaluate $\int \frac{1}{1+x^4}dx$.

I am finding it very difficult to evaluate this integral. I do not know any standard formula for this integral. I cannot think of a suitable substitution. I tried to evaluate the integral by partial fractions but it become a bit messy and integration by parts does not result in a simpler integral. How can you evaluate this integral without using partial fractions?

https://math.stackexchange.com/questions/426152/evaluating-int-0-infty-fracdx1x4 — lab bhattacharjee, Apr 09 '18 at 09:52
@BrandonLoi What you suggest won't work, because the integrand becomes, bar the mandatory considerations of sign, $\int^{x^2} \frac{1}{2(1+u^2)\sqrt u},du$ — , Apr 09 '18 at 09:56
To all users. Approach0 is often better at finding duplicates than the on-site search engine (or Google) when it is essential to include a TeX-snippet. — Jyrki Lahtonen, Apr 09 '18 at 09:59

score 1 · Answer 1 · answered Apr 09 '18 at 09:52

1

Use the fact that$$\frac1{x^4+1}=\frac1{x^4+2x^2+1-2x^2}=\frac1{\left(x^2+\sqrt2x+1\right)\left(x^2-\sqrt2x+1\right)}.$$

answered Apr 09 '18 at 09:52

José Carlos Santos

427,504

score 0 · Answer 2 · answered Apr 09 '18 at 09:51

0

Hint: $1+x^4=(1+\sqrt2x +x^2)(1-\sqrt2x +x^2)$.

answered Apr 09 '18 at 09:51

How to integrate $\frac{1}{1+x^4}$

2 Answers2