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
$\small \displaystyle \int \dfrac{1}{x^{2}-1} \,dx= \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} +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 + \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} +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 + \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} +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 + \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} +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 + \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} +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 + \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} +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= \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} + \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= \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} + \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= \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} + \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= \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} + \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= \dfrac{\log{\left(x-1 \right)}}{2}-\dfrac{\log{\left(x + 1 \right)}}{2} + \dfrac{315 x^{8} + 105 x^{6} + 63 x^{4} + 45 x^{2} + 35}{315 x^{9}} +C $