The anti-derivative of 1/(x^4+ a^4) cannot be written in terms of elementary functions with real coefficients. You could, of course, write \(\displaystyle x^4+ a^4\) as \(\displaystyle (x- \sqrt{2}/2+ \sqrt{2}{2}i)(x- \sqrt{2}/2- \sqrt{2}{2}i)(x+ \sqrt{2}/2- \sqrt{2}{2}i)(x+\sqrt{2}/2+ \sqrt{2}{2}i)\) and use "partial fractions" to write the anti-derivative as a logarithm of an expression with complex coefficients.
