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