List of integrals of inverse hyperbolic functions

Template:SHORTDESC: The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals.

• In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
• For each inverse hyperbolic integration formula below there is a corresponding formula in the list of integrals of inverse trigonometric functions.
• The ISO 80000-2 standard uses the prefix "ar-" rather than "arc-" for the inverse hyperbolic functions; we do that here.

Inverse hyperbolic sine integration formulas

$$\int \mathrm{arsinh}(ax)dx=x\mathrm{arsinh}(ax)-\frac{\sqrt{a^2{x}^2+1}}{a}+C$$

$$\int x\mathrm{arsinh}(ax)dx=\frac{x^2\mathrm{arsinh}(ax)}{2}+\frac{\mathrm{arsinh}(ax)}{4a^2}-\frac{x\sqrt{a^2{x}^2+1}}{4a}+C$$

$$\int x^2\mathrm{arsinh}(ax)dx=\frac{x^3\mathrm{arsinh}(ax)}{3}-\frac{(a^2{x}^2-2)\sqrt{a^2{x}^2+1}}{9a^3}+C$$

$$\int x^m\mathrm{arsinh}(ax)dx=\frac{x^{m+1}\mathrm{arsinh}(ax)}{m+1}-\frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{a^2{x}^2+1}}dx(m\ne -1)$$

$$\int \mathrm{arsinh}(ax{)}^{2}dx=2x+x\mathrm{arsinh}(ax{)}^2-\frac{2\sqrt{a^2{x}^2+1}\mathrm{arsinh}(ax)}{a}+C$$

$$\int \mathrm{arsinh}(ax{)}^{n}dx=x\mathrm{arsinh}(ax{)}^n-\frac{n\sqrt{a^2{x}^2+1}\mathrm{arsinh}(ax{)}^{n-1}}{a}+n(n-1)\int \mathrm{arsinh}(ax{)}^{n-2}dx$$

$$\int \mathrm{arsinh}(ax{)}^{n}dx=-\frac{x\mathrm{arsinh}(ax{)}^{n+2}}{(n+1)(n+2)}+\frac{\sqrt{a^2{x}^2+1}\mathrm{arsinh}(ax{)}^{n+1}}{a(n+1)}+\frac{1}{(n+1)(n+2)}\int \mathrm{arsinh}(ax{)}^{n+2}dx(n\ne -1,-2)$$

Inverse hyperbolic cosine integration formulas

$$\int \mathrm{arcosh}(ax)dx=x\mathrm{arcosh}(ax)-\frac{\sqrt{ax+1}\sqrt{ax-1}}{a}+C$$

$$\int x\mathrm{arcosh}(ax)dx=\frac{x^2\mathrm{arcosh}(ax)}{2}-\frac{\mathrm{arcosh}(ax)}{4a^2}-\frac{x\sqrt{ax+1}\sqrt{ax-1}}{4a}+C$$

$$\int x^2\mathrm{arcosh}(ax)dx=\frac{x^3\mathrm{arcosh}(ax)}{3}-\frac{(a^2{x}^2+2)\sqrt{ax+1}\sqrt{ax-1}}{9a^3}+C$$

$$\int x^m\mathrm{arcosh}(ax)dx=\frac{x^{m+1}\mathrm{arcosh}(ax)}{m+1}-\frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{ax+1}\sqrt{ax-1}}dx(m\ne -1)$$

$$\int \mathrm{arcosh}(ax{)}^{2}dx=2x+x\mathrm{arcosh}(ax{)}^2-\frac{2\sqrt{ax+1}\sqrt{ax-1}\mathrm{arcosh}(ax)}{a}+C$$

$$\int \mathrm{arcosh}(ax{)}^{n}dx=x\mathrm{arcosh}(ax{)}^n-\frac{n\sqrt{ax+1}\sqrt{ax-1}\mathrm{arcosh}(ax{)}^{n-1}}{a}+n(n-1)\int \mathrm{arcosh}(ax{)}^{n-2}dx$$

$$\int \mathrm{arcosh}(ax{)}^{n}dx=-\frac{x\mathrm{arcosh}(ax{)}^{n+2}}{(n+1)(n+2)}+\frac{\sqrt{ax+1}\sqrt{ax-1}\mathrm{arcosh}(ax{)}^{n+1}}{a(n+1)}+\frac{1}{(n+1)(n+2)}\int \mathrm{arcosh}(ax{)}^{n+2}dx(n\ne -1,-2)$$

Inverse hyperbolic tangent integration formulas

$$\int \mathrm{artanh}(ax)dx=x\mathrm{artanh}(ax)+\frac{\ln (1-a^2{x}^2)}{2a}+C$$

$$\int x\mathrm{artanh}(ax)dx=\frac{x^2\mathrm{artanh}(ax)}{2}-\frac{\mathrm{artanh}(ax)}{2a^2}+\frac{x}{2a}+C$$

$$\int x^2\mathrm{artanh}(ax)dx=\frac{x^3\mathrm{artanh}(ax)}{3}+\frac{\ln (1-a^2{x}^2)}{6a^3}+\frac{x^2}{6a}+C$$

$$\int x^m\mathrm{artanh}(ax)dx=\frac{x^{m+1}\mathrm{artanh}(ax)}{m+1}-\frac{a}{m+1}\int \frac{x^{m+1}}{1-a^2{x}^2}dx(m\ne -1)$$

Inverse hyperbolic cotangent integration formulas

$$\int \mathrm{arcoth}(ax)dx=x\mathrm{arcoth}(ax)+\frac{\ln (a^2{x}^2-1)}{2a}+C$$

$$\int x\mathrm{arcoth}(ax)dx=\frac{x^2\mathrm{arcoth}(ax)}{2}-\frac{\mathrm{arcoth}(ax)}{2a^2}+\frac{x}{2a}+C$$

$$\int x^2\mathrm{arcoth}(ax)dx=\frac{x^3\mathrm{arcoth}(ax)}{3}+\frac{\ln (a^2{x}^2-1)}{6a^3}+\frac{x^2}{6a}+C$$

$$\int x^m\mathrm{arcoth}(ax)dx=\frac{x^{m+1}\mathrm{arcoth}(ax)}{m+1}+\frac{a}{m+1}\int \frac{x^{m+1}}{a^2{x}^2-1}dx(m\ne -1)$$

Inverse hyperbolic secant integration formulas

$$\int \mathrm{arsech}(ax)dx=x\mathrm{arsech}(ax)-\frac{2}{a}\arctan \sqrt{\frac{1-ax}{1+ax}}+C$$

$$\int x\mathrm{arsech}(ax)dx=\frac{x^2\mathrm{arsech}(ax)}{2}-\frac{(1+ax)}{2a^2}\sqrt{\frac{1-ax}{1+ax}}+C$$

$$\int x^2\mathrm{arsech}(ax)dx=\frac{x^3\mathrm{arsech}(ax)}{3}-\frac{1}{3a^3}\arctan \sqrt{\frac{1-ax}{1+ax}}-\frac{x(1+ax)}{6a^2}\sqrt{\frac{1-ax}{1+ax}}+C$$

$$\int x^m\mathrm{arsech}(ax)dx=\frac{x^{m+1}\mathrm{arsech}(ax)}{m+1}+\frac{1}{m+1}\int \frac{x^m}{(1+ax)\sqrt{\frac{1-ax}{1+ax}}}dx(m\ne -1)$$

Inverse hyperbolic cosecant integration formulas

$$\int \mathrm{arcsch}(ax)dx=x\mathrm{arcsch}(ax)+\frac{1}{a}\mathrm{arcoth}\sqrt{\frac{1}{a^2{x}^2}+1}+C$$

$$\int x\mathrm{arcsch}(ax)dx=\frac{x^2\mathrm{arcsch}(ax)}{2}+\frac{x}{2a}\sqrt{\frac{1}{a^2{x}^2}+1}+C$$

$$\int x^2\mathrm{arcsch}(ax)dx=\frac{x^3\mathrm{arcsch}(ax)}{3}-\frac{1}{6a^3}\mathrm{arcoth}\sqrt{\frac{1}{a^2{x}^2}+1}+\frac{x^2}{6a}\sqrt{\frac{1}{a^2{x}^2}+1}+C$$

$$\int x^m\mathrm{arcsch}(ax)dx=\frac{x^{m+1}\mathrm{arcsch}(ax)}{m+1}+\frac{1}{a(m+1)}\int \frac{x^{m-1}}{\sqrt{\frac{1}{a^2{x}^2}+1}}dx(m\ne -1)$$