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