List of integrals of inverse hyperbolic functions

$$\backslash int\backslash operatorname\{arsinh\}(ax)\backslash ,dx=$$

x\operatorname{arsinh}(ax)-\frac{\sqrt{a^2x^2+1}}{a}+C

$$\backslash int\; x\backslash operatorname\{arsinh\}(ax)\backslash ,dx=$$

\frac{x^2\operatorname{arsinh}(ax)}{2}+

\frac{\operatorname{arsinh}(ax)}{4a^2}-

\frac{x \sqrt{a^2x^2+1}}{4a}+C

$$\backslash int\; x^2\backslash operatorname\{arsinh\}(ax)\backslash ,dx=$$

\frac{x^3\operatorname{arsinh}(ax)}{3}-

\frac{\left(a^2x^2-2\right)\sqrt{a^2x^2+1}}{9a^3}+C

$$\backslash int\; x^m\backslash operatorname\{arsinh\}(ax)\backslash ,dx=$$

\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)

$$\backslash int\backslash operatorname\{arsinh\}(ax)^2\backslash ,dx=$$

2x+x\operatorname{arsinh}(ax)^2-

\frac{2\sqrt{a^2x^2+1}\operatorname{arsinh}(ax)}{a}+C

$$\backslash int\backslash operatorname\{arsinh\}(ax)^n\backslash ,dx=$$

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

$$\backslash int\backslash operatorname\{arsinh\}(ax)^n\backslash ,dx=$$

-\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)

$$\backslash int\backslash operatorname\{arcosh\}(ax)\backslash ,dx=$$

x\operatorname{arcosh}(ax)-

\frac{\sqrt{ax+1}\sqrt{ax-1}}{a}+C

$$\backslash int\; x\backslash operatorname\{arcosh\}(ax)\backslash ,dx=$$

\frac{x^2\operatorname{arcosh}(ax)}{2}-

\frac{\operatorname{arcosh}(ax)}{4a^2}-

\frac{x\sqrt{ax+1}\sqrt{ax-1}}{4a}+C

$$\backslash int\; x^2\backslash operatorname\{arcosh\}(ax)\backslash ,dx=$$

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

$$\backslash int\; x^m\backslash operatorname\{arcosh\}(ax)\backslash ,dx=$$

\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)

$$\backslash int\backslash operatorname\{arcosh\}(ax)^2\backslash ,dx=$$

2x+x\operatorname{arcosh}(ax)^2-

\frac{2\sqrt{ax+1}\sqrt{ax-1}\operatorname{arcosh}(ax)}{a}+C

$$\backslash int\backslash operatorname\{arcosh\}(ax)^n\backslash ,dx=$$

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

$$\backslash int\backslash operatorname\{arcosh\}(ax)^n\backslash ,dx=$$

-\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)

$$\backslash int\backslash operatorname\{artanh\}(ax)\backslash ,dx=$$

x\operatorname{artanh}(ax)+

\frac{\ln\left(1-a^2x^2\right)}{2a}+C

$$\backslash int\; x\backslash operatorname\{artanh\}(ax)\backslash ,dx=$$

\frac{x^2\operatorname{artanh}(ax)}{2}-

\frac{\operatorname{artanh}(ax)}{2a^2}+\frac{x}{2a}+C

$$\backslash int\; x^2\backslash operatorname\{artanh\}(ax)\backslash ,dx=$$

\frac{x^3\operatorname{artanh}(ax)}{3}+

\frac{\ln\left(1-a^2x^2\right)}{6a^3}+\frac{x^2}{6a}+C

$$\backslash int\; x^m\backslash operatorname\{artanh\}(ax)\backslash ,dx=$$

\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)

$$\backslash int\backslash operatorname\{arcoth\}(ax)\backslash ,dx=$$

x\operatorname{arcoth}(ax)+

\frac{\ln\left(a^2x^2-1\right)}{2a}+C

$$\backslash int\; x\backslash operatorname\{arcoth\}(ax)\backslash ,dx=$$

\frac{x^2\operatorname{arcoth}(ax)}{2}-

\frac{\operatorname{arcoth}(ax)}{2a^2}+\frac{x}{2a}+C

$$\backslash int\; x^2\backslash operatorname\{arcoth\}(ax)\backslash ,dx=$$

\frac{x^3\operatorname{arcoth}(ax)}{3}+

\frac{\ln\left(a^2x^2-1\right)}{6a^3}+\frac{x^2}{6a}+C

$$\backslash int\; x^m\backslash operatorname\{arcoth\}(ax)\backslash ,dx=$$

\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)

$$\backslash int\backslash operatorname\{arsech\}(ax)\backslash ,dx=$$

x\operatorname{arsech}(ax)-

\frac{2}{a}\operatorname{arctan}\sqrt{\frac{1-ax}{1+ax}}+C

$$\backslash int\; x\backslash operatorname\{arsech\}(ax)\backslash ,dx=$$

\frac{x^2\operatorname{arsech}(ax)}{2}-

\frac{(1+ax)}{2a^2}\sqrt{\frac{1-ax}{1+ax}}+C

$$\backslash int\; x^2\backslash operatorname\{arsech\}(ax)\backslash ,dx=$$

\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

$$\backslash int\; x^m\backslash operatorname\{arsech\}(ax)\backslash ,dx=$$

\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)

$$\backslash int\backslash operatorname\{arcsch\}(ax)\backslash ,dx=$$

x\operatorname{arcsch}(ax)+

\frac{1}{a}\operatorname{arcoth}\sqrt{\frac{1}{a^2x^2}+1}+C

$$\backslash int\; x\backslash operatorname\{arcsch\}(ax)\backslash ,dx=$$

\frac{x^2\operatorname{arcsch}(ax)}{2}+

\frac{x}{2a}\sqrt{\frac{1}{a^2x^2}+1}+C

$$\backslash int\; x^2\backslash operatorname\{arcsch\}(ax)\backslash ,dx=$$

\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

$$\backslash int\; x^m\backslash operatorname\{arcsch\}(ax)\backslash ,dx=$$

\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)

{{Lists of integrals}}

inverse hyperbolic functions