The following is a list of integrals (antiderivative functions) of inverse hyperbolic functions. For a complete list of integral functions, see lists of integrals.
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
![{\displaystyle \int \mathrm {arsinh} \,{\frac {x}{a}}\,dx=x\,\mathrm {arsinh} \,{\frac {x}{a}}-{\sqrt {x^{2}+a^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0eff50a62e7b457823a59bb58e1760fc1f0ebf)
![{\displaystyle \int \mathrm {arcosh} \,{\frac {x}{a}}\,dx=x\,\mathrm {arcosh} \,{\frac {x}{a}}-{\sqrt {x^{2}-a^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2408c969e96492e506d031d3be47048637a94dc4)
![{\displaystyle \int \mathrm {artanh} \,{\frac {x}{a}}\,dx=x\,\mathrm {artanh} \,{\frac {x}{a}}+{\frac {a}{2}}\ln |a^{2}-x^{2}|+C\qquad {\mbox{(for }}|x|<|a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b6a829c6d9b121dafd5222600f462fec9adf8c)
![{\displaystyle \int \mathrm {arcoth} \,{\frac {x}{a}}\,dx=x\,\mathrm {arcoth} \,{\frac {x}{a}}+{\frac {a}{2}}\ln |x^{2}-a^{2}|+C\qquad {\mbox{(for }}|x|>|a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef6f07736fc306dd66822aeb69fd7549c07c0ea)
![{\displaystyle \int \mathrm {arsech} \,{\frac {x}{a}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{a}}-a\,\mathrm {arctan} \,{\frac {x\,{\sqrt {\frac {a-x}{a+x}}}}{x-a}}+C\qquad {\mbox{(for }}x\in (0,\,a){\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72e9ef87a4703beb2f724ed369615c47fe048c4b)
![{\displaystyle \int \mathrm {arcsch} \,{\frac {x}{a}}\,dx=x\,\mathrm {arcsch} \,{\frac {x}{a}}+a\,\ln \,{\frac {x+{\sqrt {x^{2}+a^{2}}}}{a}}+C\qquad {\mbox{(for }}x\in (0,\,a){\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bde600c44bfcb67b58f4ebc7a427c19dbd3b211)