Anexo:Integrales de funciones inversas hiperbólicas

La siguiente es una lista de integrales de las inversas de las funciones hiperbólicas:

∫ arsinh ⁡ x c d x = x arsinh ⁡ x c − x 2 + c 2 {\displaystyle \int \operatorname {arsinh} {\frac {x}{c}}\,dx=x\operatorname {arsinh} {\frac {x}{c}}-{\sqrt {x^{2}+c^{2}}}}
∫ arcosh ⁡ x c d x = x arcosh ⁡ x c − x 2 − c 2 {\displaystyle \int \operatorname {arcosh} {\frac {x}{c}}\,dx=x\operatorname {arcosh} {\frac {x}{c}}-{\sqrt {x^{2}-c^{2}}}}
∫ artanh ⁡ x c d x = x artanh ⁡ x c + c 2 ln ⁡ | c 2 − x 2 | ( para  | x | < | c | ) {\displaystyle \int \operatorname {artanh} {\frac {x}{c}}\,dx=x\operatorname {artanh} {\frac {x}{c}}+{\frac {c}{2}}\ln \left|c^{2}-x^{2}\right|\qquad ({\text{para }}|x|<|c|)}
∫ arcoth ⁡ x c d x = x arcoth ⁡ x c + c 2 ln ⁡ | x 2 − c 2 | ( para  | x | > | c | ) {\displaystyle \int \operatorname {arcoth} {\frac {x}{c}}\,dx=x\operatorname {arcoth} {\frac {x}{c}}+{\frac {c}{2}}\ln \left|x^{2}-c^{2}\right|\qquad ({\text{para }}|x|>|c|)}
∫ arsech ⁡ x c d x = x arsech ⁡ x c − c arctan ⁡ x c − x c + x x − c ( para  x ∈ ( 0 , c ) ) {\displaystyle \int \operatorname {arsech} {\frac {x}{c}}\,dx=x\operatorname {arsech} {\frac {x}{c}}-c\operatorname {arctan} {\frac {x\,{\sqrt {\frac {c-x}{c+x}}}}{x-c}}\qquad \left({\text{para }}x\in (0,c)\right)}
∫ arcsch ⁡ x c d x = x arcsch ⁡ x c + c ln x + x 2 + c 2 c ( para  x ∈ ( 0 , c ) ) {\displaystyle \int \operatorname {arcsch} {\frac {x}{c}}\,dx=x\operatorname {arcsch} {\frac {x}{c}}+c\,\ln \,{\frac {x+{\sqrt {x^{2}+c^{2}}}}{c}}\qquad \left({\text{para }}x\in (0,c)\right)}