不定積分 $\small \displaystyle \int \dfrac{x^{n}}{x^{4}+1} \,dx$ の結果をまとめています。
$n \geqq 0$ の場合
（※$\operatorname{atan}$は正接（タンジェント）の逆関数）
$\small \displaystyle \int \dfrac{1}{x^{4} + 1} \,dx$ $\small =-\dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8} + \dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} $ $\quad \small + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4} + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4} +C $
$\small \displaystyle \int \dfrac{x}{x^{4} + 1} \,dx$ $\small = \dfrac{\operatorname{atan}{\left(x^{2} \right)}}{2} +C $
$\small \displaystyle \int \dfrac{x^{2}}{x^{4} + 1} \,dx$ $\small = \dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8}-\dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} $ $\quad \small + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4} + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4} +C $
$\small \displaystyle \int \dfrac{x^{3}}{x^{4} + 1} \,dx$ $\small = \dfrac{\log{\left(x^{4} + 1 \right)}}{4} +C $
$\small \displaystyle \int \dfrac{x^{4}}{x^{4} + 1} \,dx$ $\small = x + \dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8}-\dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} $ $\quad \small -\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4}-\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4} +C $
$\small \displaystyle \int \dfrac{x^{5}}{x^{4} + 1} \,dx$ $\small = \dfrac{x^{2}}{2}-\dfrac{\operatorname{atan}{\left(x^{2} \right)}}{2} +C $
$\small \displaystyle \int \dfrac{x^{6}}{x^{4} + 1} \,dx$ $\small = \dfrac{x^{3}}{3}-\dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8} + \dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} $ $\quad \small -\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4}-\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4} +C $
$\small \displaystyle \int \dfrac{x^{7}}{x^{4} + 1} \,dx$ $\small = \dfrac{x^{4}}{4}-\dfrac{\log{\left(x^{4} + 1 \right)}}{4} +C $
$\small \displaystyle \int \dfrac{x^{8}}{x^{4} + 1} \,dx$ $\small = \dfrac{x^{5}}{5}-x-\dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8} + \dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8}$ $\quad \small + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4} + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4} +C $
$\small \displaystyle \int \dfrac{x^{9}}{x^{4} + 1} \,dx$ $\small = \dfrac{x^{6}}{6}-\dfrac{x^{2}}{2} + \dfrac{\operatorname{atan}{\left(x^{2} \right)}}{2} +C $
$\small \displaystyle \int \dfrac{x^{10}}{x^{4} + 1} \,dx$ $\small = \dfrac{x^{7}}{7}-\dfrac{x^{3}}{3} + \dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8}-\dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8}$ $\quad \small + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4} + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4} +C $
$n < 0$ の場合
$\small \displaystyle \int \dfrac{1}{x \left(x^{4} + 1\right)} \,dx$ $\small = \log{\left(x \right)}-\dfrac{\log{\left(x^{4} + 1 \right)}}{4} +C $
$\small \displaystyle \int \dfrac{1}{x^{2} \left(x^{4} + 1\right)} \,dx$ $\small =-\dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8} + \dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8}$ $\quad \small -\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4}-\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4}-\dfrac{1}{x} +C $
$\small \displaystyle \int \dfrac{1}{x^{3} \left(x^{4} + 1\right)} \,dx$ $\small =-\dfrac{\operatorname{atan}{\left(x^{2} \right)}}{2}-\dfrac{1}{2 x^{2}} +C $
$\small \displaystyle \int \dfrac{1}{x^{4} \left(x^{4} + 1\right)} \,dx$ $\small = \dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8}-\dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8}$ $\quad \small -\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4}-\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4}-\dfrac{1}{3 x^{3}} +C $
$\small \displaystyle \int \dfrac{1}{x^{5} \left(x^{4} + 1\right)} \,dx$ $\small =-\log{\left(x \right)} + \dfrac{\log{\left(x^{4} + 1 \right)}}{4}-\dfrac{1}{4 x^{4}} +C $
$\small \displaystyle \int \dfrac{1}{x^{6} \left(x^{4} + 1\right)} \,dx$ $\small = \dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8}-\dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} $ $\quad \small + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4} + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4} + \dfrac{5 x^{4}-1}{5 x^{5}} +C $
$\small \displaystyle \int \dfrac{1}{x^{7} \left(x^{4} + 1\right)} \,dx$ $\small = \dfrac{\operatorname{atan}{\left(x^{2} \right)}}{2} + \dfrac{3 x^{4}-1}{6 x^{6}} +C $
$\small \displaystyle \int \dfrac{1}{x^{8} \left(x^{4} + 1\right)} \,dx$ $\small =-\dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8} + \dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} $ $\quad \small + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4} + \dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4} + \dfrac{7 x^{4}-3}{21 x^{7}} +C $
$\small \displaystyle \int \dfrac{1}{x^{9} \left(x^{4} + 1\right)} \,dx$ $\small = \log{\left(x \right)}-\dfrac{\log{\left(x^{4} + 1 \right)}}{4} + \dfrac{2 x^{4}-1}{8 x^{8}} +C $
$\small \displaystyle \int \dfrac{1}{x^{10} \left(x^{4} + 1\right)} \,dx$ $\small =-\dfrac{\sqrt{2} \log{\left(x^{2}-\sqrt{2} x + 1 \right)}}{8} + \dfrac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8}$ $\quad \small -\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x-1 \right)}}{4}-\dfrac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4}-\dfrac{45 x^{8}-9 x^{4} + 5}{45 x^{9}} +C $