不定積分 $\small \displaystyle \int \dfrac{x^{n}}{x^{2}+1} \,dx$ 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^{2} + 1} \,dx= \operatorname{atan}{\left(x \right)} +C $
$\small \displaystyle \int \dfrac{x}{x^{2} + 1} \,dx= \dfrac{\log{\left(x^{2} + 1 \right)}}{2} +C $
$\small \displaystyle \int \dfrac{x^{2}}{x^{2} + 1} \,dx= x-\operatorname{atan}{\left(x \right)} +C $
$\small \displaystyle \int \dfrac{x^{3}}{x^{2} + 1} \,dx= \dfrac{x^{2}}{2}-\dfrac{\log{\left(x^{2} + 1 \right)}}{2} +C $
$\small \displaystyle \int \dfrac{x^{4}}{x^{2} + 1} \,dx= \dfrac{x^{3}}{3}-x + \operatorname{atan}{\left(x \right)} +C $
$\small \displaystyle \int \dfrac{x^{5}}{x^{2} + 1} \,dx= \dfrac{x^{4}}{4}-\dfrac{x^{2}}{2} + \dfrac{\log{\left(x^{2} + 1 \right)}}{2} +C $
$\small \displaystyle \int \dfrac{x^{6}}{x^{2} + 1} \,dx= \dfrac{x^{5}}{5}-\dfrac{x^{3}}{3} + x-\operatorname{atan}{\left(x \right)} +C $
$\small \displaystyle \int \dfrac{x^{7}}{x^{2} + 1} \,dx= \dfrac{x^{6}}{6}-\dfrac{x^{4}}{4} + \dfrac{x^{2}}{2}-\dfrac{\log{\left(x^{2} + 1 \right)}}{2} +C $
$\small \displaystyle \int \dfrac{x^{8}}{x^{2} + 1} \,dx= \dfrac{x^{7}}{7}-\dfrac{x^{5}}{5} + \dfrac{x^{3}}{3}-x + \operatorname{atan}{\left(x \right)} +C $
$\small \displaystyle \int \dfrac{x^{9}}{x^{2} + 1} \,dx= \dfrac{x^{8}}{8}-\dfrac{x^{6}}{6} + \dfrac{x^{4}}{4}-\dfrac{x^{2}}{2} + \dfrac{\log{\left(x^{2} + 1 \right)}}{2} +C $
$\small \displaystyle \int \dfrac{x^{10}}{x^{2} + 1} \,dx= \dfrac{x^{9}}{9}-\dfrac{x^{7}}{7} + \dfrac{x^{5}}{5}-\dfrac{x^{3}}{3} + x-\operatorname{atan}{\left(x \right)} +C $
$n < 0$ の場合
$\small \displaystyle \int \dfrac{1}{x \left(x^{2} + 1\right)} \,dx= \log{\left(x \right)}-\dfrac{\log{\left(x^{2} + 1 \right)}}{2} +C $
$\small \displaystyle \int \dfrac{1}{x^{2} \left(x^{2} + 1\right)} \,dx=-\operatorname{atan}{\left(x \right)}-\dfrac{1}{x} +C $
$\small \displaystyle \int \dfrac{1}{x^{3} \left(x^{2} + 1\right)} \,dx=-\log{\left(x \right)} + \dfrac{\log{\left(x^{2} + 1 \right)}}{2}-\dfrac{1}{2 x^{2}} +C $
$\small \displaystyle \int \dfrac{1}{x^{4} \left(x^{2} + 1\right)} \,dx= \operatorname{atan}{\left(x \right)} + \dfrac{3 x^{2}-1}{3 x^{3}} +C $
$\small \displaystyle \int \dfrac{1}{x^{5} \left(x^{2} + 1\right)} \,dx= \log{\left(x \right)}-\dfrac{\log{\left(x^{2} + 1 \right)}}{2} + \dfrac{2 x^{2}-1}{4 x^{4}} +C $
$\small \displaystyle \int \dfrac{1}{x^{6} \left(x^{2} + 1\right)} \,dx=-\operatorname{atan}{\left(x \right)}-\dfrac{15 x^{4}-5 x^{2} + 3}{15 x^{5}} +C $
$\small \displaystyle \int \dfrac{1}{x^{7} \left(x^{2} + 1\right)} \,dx=-\log{\left(x \right)} + \dfrac{\log{\left(x^{2} + 1 \right)}}{2}-\dfrac{6 x^{4}-3 x^{2} + 2}{12 x^{6}} +C $
$\small \displaystyle \int \dfrac{1}{x^{8} \left(x^{2} + 1\right)} \,dx= \operatorname{atan}{\left(x \right)} + \dfrac{105 x^{6}-35 x^{4} + 21 x^{2}-15}{105 x^{7}} +C $
$\small \displaystyle \int \dfrac{1}{x^{9} \left(x^{2} + 1\right)} \,dx= \log{\left(x \right)}-\dfrac{\log{\left(x^{2} + 1 \right)}}{2} + \dfrac{12 x^{6}-6 x^{4} + 4 x^{2}-3}{24 x^{8}} +C $
$\small \displaystyle \int \dfrac{1}{x^{10} \left(x^{2} + 1\right)} \,dx=-\operatorname{atan}{\left(x \right)}-\dfrac{315 x^{8}-105 x^{6} + 63 x^{4}-45 x^{2} + 35}{315 x^{9}} +C $