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.

Inverse hyperbolic sine integration formulas

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

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

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

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

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

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

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

Inverse hyperbolic cosine integration formulas

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

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

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

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

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

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

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

Inverse hyperbolic tangent integration formulas

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

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

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

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

Inverse hyperbolic cotangent integration formulas

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

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

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

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

Inverse hyperbolic secant integration formulas

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

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

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

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

Inverse hyperbolic cosecant integration formulas

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

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

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

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