逆双曲線関数の原始関数の一覧
∫ x 2 arcoth  ( a x ) d x = x 3 arcoth  ( a x ) 3 + ln  ( a 2 x 2 − 1 ) 6 a 3 + x 2 6 a + C {\displaystyle \int x^{2}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{3}\,\operatorname {arcoth} (a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C}
∫ x m arcoth  ( a x ) d x = x m + 1 arcoth  ( a x ) m + 1 + a m + 1 ∫ x m + 1 a 2 x 2 − 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{m+1}\operatorname {arcoth} (a\,x)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}-1}}\,dx\quad (m\neq -1)}