逆双曲線関数の原始関数の一覧
∫ arcosh  ( a x ) d x = x arcosh  ( a x ) − a x + 1 a x − 1 a + C {\displaystyle \int \operatorname {arcosh} (a\,x)\,dx=x\,\operatorname {arcosh} (a\,x)-{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{a}}+C}
∫ x arcosh  ( a x ) d x = x 2 arcosh  ( a x ) 2 − arcosh  ( a x ) 4 a 2 − x a x + 1 a x − 1 4 a + C {\displaystyle \int x\,\operatorname {arcosh} (a\,x)dx={\frac {x^{2}\,\operatorname {arcosh} (a\,x)}{2}}-{\frac {\operatorname {arcosh} (a\,x)}{4\,a^{2}}}-{\frac {x\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{4\,a}}+C}
∫ x m arsinh  ( a x ) d x = x m + 1 arsinh  ( a x ) m + 1 − a m + 1 ∫ x m + 1 a 2 x 2 + 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsinh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {a^{2}\,x^{2}+1}}}\,dx\quad (m\neq -1)}
∫ arsinh  ( a x ) 2 d x = 2 x + x arsinh  ( a x ) 2 − 2 a 2 x 2 + 1 arsinh  ( a x ) a + C {\displaystyle \int \operatorname {arsinh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arsinh} (a\,x)^{2}-{\frac {2\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)}{a}}+C}
∫ x 2 arcosh  ( a x ) d x = x 3 arcosh  ( a x ) 3 − ( a 2 x 2 + 2 ) a x + 1 a x − 1 9 a 3 + C {\displaystyle \int x^{2}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{3}\,\operatorname {arcosh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}+2\right){\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{9\,a^{3}}}+C}
∫ x m arcosh  ( a x ) d x = x m + 1 arcosh  ( a x ) m + 1 − a m + 1 ∫ x m + 1 a x + 1 a x − 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arcosh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}}\,dx\quad (m\neq -1)}
∫ x 2 arsech  ( a x ) d x = x 3 arsech  ( a x ) 3 − 1 3 a 3 arctan  1 − a x 1 + a x − x ( 1 + a x ) 6 a 2 1 − a x 1 + a x + C {\displaystyle \int x^{2}\,\operatorname {arsech} (a\,x)dx={\frac {x^{3}\,\operatorname {arsech} (a\,x)}{3}}\,-\,{\frac {1}{3\,a^{3}}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}\,-\,{\frac {x(1+a\,x)}{6\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}\,+\,C}
∫ x m arsech  ( a x ) d x = x m + 1 arsech  ( a x ) m + 1 + 1 m + 1 ∫ x m ( 1 + a x ) 1 − a x 1 + a x d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arsech} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsech} (a\,x)}{m+1}}\,+\,{\frac {1}{m+1}}\int {\frac {x^{m}}{(1+a\,x){\sqrt {\frac {1-a\,x}{1+a\,x}}}}}\,dx\quad (m\neq -1)}