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