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