List of integrals of inverse hyperbolic functions

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

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

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

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

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

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

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

Inverse hyperbolic cosine integration formulas
$$\int\operatorname{arcosh}(ax)\,dx= x\operatorname{arcosh}(ax)- \frac{\sqrt{ax+1}\sqrt{ax-1}}{a}+C$$

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

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

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

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

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

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

Inverse hyperbolic tangent integration formulas
$$\int\operatorname{artanh}(ax)\,dx= x\operatorname{artanh}(ax)+ \frac{\ln\left(1-a^2x^2\right)}{2a}+C$$

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

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

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

Inverse hyperbolic cotangent integration formulas
$$\int\operatorname{arcoth}(ax)\,dx= x\operatorname{arcoth}(ax)+ \frac{\ln\left(a^2x^2-1\right)}{2a}+C$$

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

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

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

Inverse hyperbolic secant integration formulas
$$\int\operatorname{arsech}(ax)\,dx= x\operatorname{arsech}(ax)- \frac{2}{a}\operatorname{arctan}\sqrt{\frac{1-ax}{1+ax}}+C$$

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

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

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

Inverse hyperbolic cosecant integration formulas
$$\int\operatorname{arcsch}(ax)\,dx= x\operatorname{arcsch}(ax)+ \frac{1}{a}\operatorname{arcoth}\sqrt{\frac{1}{a^2x^2}+1}+C$$

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

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

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