Integrals Involving $\cosh ax$

$\displaystyle \int \cosh ax\,dx = \frac{\sinh ax}{a}$

$\displaystyle \int x\cosh ax\,dx = \frac{x\sinh ax}{a} - \frac{\cosh ax}{a^2}$

$\displaystyle \int x^2\cosh ax\,dx = -\frac{2x\cosh ax}{a^2} + \left(\frac{x^2}{a} + \frac{2}{a^3}\right)\sinh ax$

$\displaystyle \int \frac{\cosh ax}{x}\,dx = \ln x + \frac{(ax)^2}{2\cdot 2!} + \frac{(ax)^4}{4\cdot 4!} + \cdots$

$\displaystyle \int \frac{\cosh ax}{x^2}\,dx = -\frac{\cosh ax}{x} + a\int \frac{\sinh ax}{x}\,dx$

$\displaystyle \int \frac{dx}{\cosh ax} = \frac{2}{a}\tan^{-1}e^{ax}$

$\displaystyle \int \frac{x\,dx}{\cosh ax} = \frac{1}{a^2}\bigg\{\frac{(ax)^2}{2} - \frac{(ax)^4}{8} + \frac{5(ax)^6}{144} - \cdots + \frac{(-1)^nE_n(ax)^{2n+2}}{(2n + 2)(2n)!} + \cdots\bigg\}$

Here $E_n$ are the Euler numbers: $E_0 = 1$, $E_1 = 1$, $E_2 = 5$, $E_3 = 61$, $E_4 = 1385, \ldots$

$\displaystyle \int \cosh^2 ax\,dx = \frac{x}{2} + \frac{\sinh 2ax}{4a}$

$\displaystyle \int x\cosh^2 ax\,dx = \frac{x^2}{4} + \frac{x\sinh 2ax}{4a} - \frac{\cosh 2ax}{8a^2}$

$\displaystyle \int \frac{dx}{\cosh^2 ax} = \frac{\tanh ax}{a}$

$\displaystyle \int \cosh ax\cosh px\,dx = \frac{\sinh(a - p)x}{2(a - p)} + \frac{\sinh(a + p)x}{2(a + p)}$

$\displaystyle \int \cosh ax\sin px\,dx = \frac{a\sinh ax\sin px - p\cosh ax\cos px}{a^2 + p^2}$

$\displaystyle \int \cosh ax\cos px\,dx = \frac{a\sinh ax\cos px + p\cosh ax\sin px}{a^2 + p^2}$

$\displaystyle \int \frac{dx}{\cosh ax + 1} = \frac{1}{a}\tanh\frac{ax}{2}$

$\displaystyle \int \frac{dx}{\cosh ax - 1} = -\frac{1}{a}\coth\frac{ax}{2}$

$\displaystyle \int \frac{x\,dx}{\cosh ax + 1} = \frac{x}{a}\tanh\frac{ax}{2} - \frac{2}{a^2}\ln\cosh\frac{ax}{2}$

$\displaystyle \int \frac{x\,dx}{\cosh ax - 1} = -\frac{x}{a}\coth\frac{ax}{2} + \frac{2}{a^2}\ln\sinh\frac{ax}{2}$

$\displaystyle \int \frac{dx}{(\cosh ax + 1)^2} = \frac{1}{2a}\tanh\frac{ax}{2} - \frac{1}{6a}\tanh^3\frac{ax}{2}$

$\displaystyle \int \frac{dx}{(\cosh ax - 1)^2} = \frac{1}{2a}\coth\frac{ax}{2} - \frac{1}{6a}\coth^3\frac{ax}{2}$

$\displaystyle \int \frac{dx}{p + q\cosh ax} = \frac{2}{a\sqrt{q^2 - p^2}}\tan^{-1}\frac{qe^{ax} + p}{\sqrt{q^2 - p^2}}$ $[q^2 > p^2]$

$\displaystyle \int \frac{dx}{p + q\cosh ax} = \frac{1}{a\sqrt{p^2 - q^2}}\,\allowbreak\ln\left(\frac{qe^{ax} + p - \sqrt{p^2 - q^2}}{qe^{ax} + p + \sqrt{p^2 - q^2}}\right)$ $[p^2 > q^2]$

$\displaystyle \int \frac{dx}{(p + q\cosh ax)^2} = \frac{q\sinh ax}{a(q^2 - p^2)(p + q\cosh ax)} - \frac{p}{q^2 - p^2}\int \frac{dx}{p + q\cosh ax}$

$\displaystyle \int \frac{dx}{p^2 - q^2\cosh^2 ax} = \frac{1}{2ap\sqrt{p^2 - q^2}}\,\allowbreak\ln\left(\frac{p\tanh ax + \sqrt{p^2 - q^2}}{p\tanh ax - \sqrt{p^2 - q^2}}\right)$ $[p^2 > q^2]$

$\displaystyle \int \frac{dx}{p^2 - q^2\cosh^2 ax} = -\frac{1}{ap\sqrt{q^2 - p^2}}\tan^{-1}\frac{p\tanh ax}{\sqrt{q^2 - p^2}}$ $[q^2 > p^2]$

$\displaystyle \int \frac{dx}{p^2 + q^2\cosh^2 ax} = \frac{1}{2ap\sqrt{p^2 + q^2}}\,\allowbreak\ln\left(\frac{\sqrt{p^2 + q^2} + p\tanh ax}{\sqrt{p^2 + q^2} - p\tanh ax}\right)$

$\displaystyle \int x^m\cosh ax\,dx = \frac{x^m\sinh ax}{a} - \frac{m}{a}\int x^{m-1}\sinh ax\,dx$

$\displaystyle \int \cosh^n ax\,dx = \frac{\cosh^{n-1} ax\sinh ax}{an} + \frac{n - 1}{n}\int \cosh^{n-2} ax\,dx$

$\displaystyle \int \frac{\cosh ax}{x^n}\,dx = -\frac{\cosh ax}{(n - 1)x^{n-1}} + \frac{a}{n - 1}\int \frac{\sinh ax}{x^{n-1}}\,dx$

$\displaystyle \int \frac{dx}{\cosh^n ax} = \frac{\sinh ax}{a(n - 1)\cosh^{n-1} ax} + \frac{n - 2}{n - 1}\int \frac{dx}{\cosh^{n-2} ax}$

$\displaystyle \int \frac{x\,dx}{\cosh^n ax} = \frac{x\sinh ax}{a(n - 1)\cosh^{n-1} ax} + \frac{1}{(n - 1)(n - 2)a^2\cosh^{n-2} ax} + \frac{n - 2}{n - 1}\int \frac{x\,dx}{\cosh^{n-2} ax}$

Integrals Involving $\sinh ax$ and $\cosh ax$

$\displaystyle \int \sinh ax\cosh ax\,dx = \frac{\sinh^2 ax}{2a}$

$\displaystyle \int \sinh px\cosh qx\,dx = \frac{\cosh(p + q)x}{2(p + q)} + \frac{\cosh(p - q)x}{2(p - q)}$

$\displaystyle \int \sinh^n ax\cosh ax\,dx = \frac{\sinh^{n+1} ax}{(n + 1)a}$

$\displaystyle \int \cosh^n ax\sinh ax\,dx = \frac{\cosh^{n+1} ax}{(n + 1)a}$

$\displaystyle \int \sinh^2 ax\cosh^2 ax\,dx = \frac{\sinh 4ax}{32a} - \frac{x}{8}$

$\displaystyle \int \frac{dx}{\sinh ax\cosh ax} = \frac{1}{a}\ln(\tanh ax)$

$\displaystyle \int \frac{dx}{\sinh^2 ax\cosh ax} = -\frac{1}{a}\tan^{-1}(\sinh ax) - \frac{\operatorname{csch} ax}{a}$

$\displaystyle \int \frac{dx}{\sinh ax\cosh^2 ax} = \frac{\operatorname{sech} ax}{a} + \frac{1}{a}\ln\tanh\frac{ax}{2}$

$\displaystyle \int \frac{dx}{\sinh^2 ax\cosh^2 ax} = -\frac{2\coth 2ax}{a}$

$\displaystyle \int \frac{\sinh^2 ax}{\cosh ax}\,dx = \frac{\sinh ax}{a} - \frac{1}{a}\tan^{-1}(\sinh ax)$

$\displaystyle \int \frac{\cosh^2 ax}{\sinh ax}\,dx = \frac{\cosh ax}{a} + \frac{1}{a}\ln\tanh\frac{ax}{2}$

$\displaystyle \int \frac{dx}{\cosh ax\,(1 + \sinh ax)} = \frac{1}{2a}\ln\left(\frac{1 + \sinh ax}{\cosh ax}\right) + \frac{1}{a}\tan^{-1}e^{ax}$

$\displaystyle \int \frac{dx}{\sinh ax\,(\cosh ax + 1)} = \frac{1}{2a}\ln\tanh\frac{ax}{2} + \frac{1}{2a(\cosh ax + 1)}$

$\displaystyle \int \frac{dx}{\sinh ax\,(\cosh ax - 1)} = -\frac{1}{2a}\ln\tanh\frac{ax}{2} - \frac{1}{2a(\cosh ax - 1)}$

Integrals Involving $\tanh ax$

$\displaystyle \int \tanh ax\,dx = \frac{1}{a}\ln(\cosh ax)$

$\displaystyle \int \tanh^2 ax\,dx = x - \frac{\tanh ax}{a}$

$\displaystyle \int \tanh^3 ax\,dx = \frac{1}{a}\ln(\cosh ax) - \frac{\tanh^2 ax}{2a}$

$\displaystyle \int \tanh^n ax\operatorname{sech}^2 ax\,dx = \frac{\tanh^{n+1} ax}{(n + 1)a}$

$\displaystyle \int \frac{\operatorname{sech}^2 ax}{\tanh ax}\,dx = \frac{1}{a}\ln(\tanh ax)$

$\displaystyle \int \frac{dx}{\tanh ax} = \frac{1}{a}\ln(\sinh ax)$

$\displaystyle \int x\tanh ax\,dx = \frac{1}{a^2}\bigg\{\frac{(ax)^3}{3} - \frac{(ax)^5}{15} + \frac{2(ax)^7}{105} - \cdots + \frac{(-1)^{n-1}2^{2n}(2^{2n} - 1)B_n(ax)^{2n+1}}{(2n + 1)!} + \cdots\bigg\}$

Here $B_n$ are the Bernoulli numbers: $B_1 = \frac{1}{6}$, $B_2 = \frac{1}{30}$, $B_3 = \frac{1}{42}$, $B_4 = \frac{1}{30}$, $B_5 = \frac{5}{66}, \ldots$

$\displaystyle \int x\tanh^2 ax\,dx = \frac{x^2}{2} - \frac{x\tanh ax}{a} + \frac{1}{a^2}\ln(\cosh ax)$

$\displaystyle \int \frac{\tanh ax}{x}\,dx = ax - \frac{(ax)^3}{9} + \frac{2(ax)^5}{75} - \cdots + \frac{(-1)^{n-1}2^{2n}(2^{2n} - 1)B_n(ax)^{2n-1}}{(2n - 1)(2n)!} + \cdots$

$\displaystyle \int \frac{dx}{p + q\tanh ax} = \frac{px}{p^2 - q^2} - \frac{q}{a(p^2 - q^2)}\,\allowbreak\ln(q\sinh ax +\nobreak p\cosh ax)$

$\displaystyle \int \tanh^n ax\,dx = -\frac{\tanh^{n-1} ax}{a(n - 1)} + \int \tanh^{n-2} ax\,dx$

Integrals Involving $\coth ax$

$\displaystyle \int \coth ax\,dx = \frac{1}{a}\ln(\sinh ax)$

$\displaystyle \int \coth^2 ax\,dx = x - \frac{\coth ax}{a}$

$\displaystyle \int \coth^3 ax\,dx = \frac{1}{a}\ln(\sinh ax) - \frac{\coth^2 ax}{2a}$

$\displaystyle \int \coth^n ax\operatorname{csch}^2 ax\,dx = -\frac{\coth^{n+1} ax}{(n + 1)a}$

$\displaystyle \int \frac{\operatorname{csch}^2 ax}{\coth ax}\,dx = -\frac{1}{a}\ln(\coth ax)$

$\displaystyle \int \frac{dx}{\coth ax} = \frac{1}{a}\ln(\cosh ax)$

$\displaystyle \int x\coth ax\,dx = \frac{1}{a^2}\bigg\{ax + \frac{(ax)^3}{9} - \frac{(ax)^5}{225} + \cdots + \frac{(-1)^{n-1}2^{2n}B_n(ax)^{2n+1}}{(2n + 1)!} + \cdots\bigg\}$

$\displaystyle \int x\coth^2 ax\,dx = \frac{x^2}{2} - \frac{x\coth ax}{a} + \frac{1}{a^2}\ln(\sinh ax)$

$\displaystyle \int \frac{\coth ax}{x}\,dx = -\frac{1}{ax} + \frac{ax}{3} - \frac{(ax)^3}{135} + \cdots + \frac{(-1)^{n-1}2^{2n}B_n(ax)^{2n-1}}{(2n - 1)(2n)!} + \cdots$

$\displaystyle \int \frac{dx}{p + q\coth ax} = \frac{px}{p^2 - q^2} - \frac{q}{a(p^2 - q^2)}\,\allowbreak\ln(p\sinh ax +\nobreak q\cosh ax)$

$\displaystyle \int \coth^n ax\,dx = -\frac{\coth^{n-1} ax}{a(n - 1)} + \int \coth^{n-2} ax\,dx$

Integrals Involving $\operatorname{sech} ax$

$\displaystyle \int \operatorname{sech} ax\,dx = \frac{2}{a}\tan^{-1}e^{ax}$

$\displaystyle \int \operatorname{sech}^2 ax\,dx = \frac{\tanh ax}{a}$

$\displaystyle \int \operatorname{sech}^3 ax\,dx = \frac{\operatorname{sech} ax\tanh ax}{2a} + \frac{1}{2a}\tan^{-1}(\sinh ax)$

$\displaystyle \int \operatorname{sech}^n ax\tanh ax\,dx = -\frac{\operatorname{sech}^n ax}{na}$

$\displaystyle \int \frac{dx}{\operatorname{sech} ax} = \frac{\sinh ax}{a}$

$\displaystyle \int x\operatorname{sech} ax\,dx = \frac{1}{a^2}\bigg\{\frac{(ax)^2}{2} - \frac{(ax)^4}{8} + \frac{5(ax)^6}{144} - \cdots + \frac{(-1)^nE_n(ax)^{2n+2}}{(2n + 2)(2n)!} + \cdots\bigg\}$

$\displaystyle \int x\operatorname{sech}^2 ax\,dx = \frac{x\tanh ax}{a} - \frac{1}{a^2}\ln(\cosh ax)$

$\displaystyle \int \frac{\operatorname{sech} ax}{x}\,dx = \ln x - \frac{(ax)^2}{4} + \frac{5(ax)^4}{96} - \frac{61(ax)^6}{4320} + \cdots + \frac{(-1)^nE_n(ax)^{2n}}{2n(2n)!} + \cdots$

$\displaystyle \int \frac{dx}{q + p\operatorname{sech} ax} = \frac{x}{q} - \frac{p}{q}\int \frac{dx}{p + q\cosh ax}$

$\displaystyle \int \operatorname{sech}^n ax\,dx = \frac{\operatorname{sech}^{n-2} ax\tanh ax}{a(n - 1)} + \frac{n - 2}{n - 1}\int \operatorname{sech}^{n-2} ax\,dx$

Integrals Involving $\operatorname{csch} ax$

$\displaystyle \int \operatorname{csch} ax\,dx = \frac{1}{a}\ln\tanh\frac{ax}{2}$

$\displaystyle \int \operatorname{csch}^2 ax\,dx = -\frac{\coth ax}{a}$

$\displaystyle \int \operatorname{csch}^3 ax\,dx = -\frac{\operatorname{csch} ax\coth ax}{2a} - \frac{1}{2a}\ln\tanh\frac{ax}{2}$

$\displaystyle \int \operatorname{csch}^n ax\coth ax\,dx = -\frac{\operatorname{csch}^n ax}{na}$

$\displaystyle \int \frac{dx}{\operatorname{csch} ax} = \frac{\cosh ax}{a}$

$\displaystyle \int x\operatorname{csch} ax\,dx = \frac{1}{a^2}\bigg\{ax - \frac{(ax)^3}{18} + \frac{7(ax)^5}{1800} - \cdots + \frac{2(-1)^n(2^{2n-1} - 1)B_n(ax)^{2n+1}}{(2n + 1)!} + \cdots\bigg\}$

$\displaystyle \int x\operatorname{csch}^2 ax\,dx = -\frac{x\coth ax}{a} + \frac{1}{a^2}\ln(\sinh ax)$

$\displaystyle \int \frac{\operatorname{csch} ax}{x}\,dx = -\frac{1}{ax} - \frac{ax}{6} + \frac{7(ax)^3}{1080} - \cdots + \frac{2(-1)^n(2^{2n-1} - 1)B_n(ax)^{2n-1}}{(2n - 1)(2n)!} + \cdots$

$\displaystyle \int \frac{dx}{q + p\operatorname{csch} ax} = \frac{x}{q} - \frac{p}{q}\int \frac{dx}{p + q\sinh ax}$

$\displaystyle \int \operatorname{csch}^n ax\,dx = -\frac{\operatorname{csch}^{n-2} ax\coth ax}{a(n - 1)} - \frac{n - 2}{n - 1}\int \operatorname{csch}^{n-2} ax\,dx$

Integrals Involving Inverse Hyperbolic Functions

$\displaystyle \int \sinh^{-1}\frac{x}{a}\,dx = x\sinh^{-1}\frac{x}{a} - \sqrt{x^2 + a^2}$

$\displaystyle \int x\sinh^{-1}\frac{x}{a}\,dx = \left(\frac{x^2}{2} + \frac{a^2}{4}\right)\sinh^{-1}\frac{x}{a} - \frac{x\sqrt{x^2 + a^2}}{4}$

$\displaystyle \int x^2\sinh^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\sinh^{-1}\frac{x}{a} + \frac{(2a^2 - x^2)\sqrt{x^2 + a^2}}{9}$

$\displaystyle \int \frac{\sinh^{-1}(x/a)}{x}\,dx = \frac{x}{a} - \frac{(x/a)^3}{2\cdot 3\cdot 3} + \frac{1\cdot 3\,(x/a)^5}{2\cdot 4\cdot 5\cdot 5} - \frac{1\cdot 3\cdot 5\,(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdots$ $[|x| \lt a]$

$\displaystyle \int \frac{\sinh^{-1}(x/a)}{x}\,dx = \frac{\ln^2(2x/a)}{2} - \frac{(a/x)^2}{2\cdot 2\cdot 2} + \frac{1\cdot 3\,(a/x)^4}{2\cdot 4\cdot 4\cdot 4} - \frac{1\cdot 3\cdot 5\,(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6} + \cdots$ $[x > a]$

$\displaystyle \int \frac{\sinh^{-1}(x/a)}{x}\,dx = -\frac{\ln^2(-2x/a)}{2} + \frac{(a/x)^2}{2\cdot 2\cdot 2} - \frac{1\cdot 3\,(a/x)^4}{2\cdot 4\cdot 4\cdot 4} + \frac{1\cdot 3\cdot 5\,(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6} - \cdots$ $[x \lt -a]$

$\displaystyle \int \frac{\sinh^{-1}(x/a)}{x^2}\,dx = -\frac{\sinh^{-1}(x/a)}{x} - \frac{1}{a}\ln\left(\frac{a + \sqrt{x^2 + a^2}}{x}\right)$

$\displaystyle \int \cosh^{-1}\frac{x}{a}\,dx = x\cosh^{-1}\frac{x}{a} - \sqrt{x^2 - a^2}$ $[\cosh^{-1}(x/a) > 0]$

$\displaystyle \int \cosh^{-1}\frac{x}{a}\,dx = x\cosh^{-1}\frac{x}{a} + \sqrt{x^2 - a^2}$ $[\cosh^{-1}(x/a) \lt 0]$

$\displaystyle \int x\cosh^{-1}\frac{x}{a}\,dx = \frac{2x^2 - a^2}{4}\cosh^{-1}\frac{x}{a} - \frac{x\sqrt{x^2 - a^2}}{4}$ $[\cosh^{-1}(x/a) > 0]$

$\displaystyle \int x\cosh^{-1}\frac{x}{a}\,dx = \frac{2x^2 - a^2}{4}\cosh^{-1}\frac{x}{a} + \frac{x\sqrt{x^2 - a^2}}{4}$ $[\cosh^{-1}(x/a) \lt 0]$

$\displaystyle \int x^2\cosh^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\cosh^{-1}\frac{x}{a} - \frac{(x^2 + 2a^2)\sqrt{x^2 - a^2}}{9}$ $[\cosh^{-1}(x/a) > 0]$

$\displaystyle \int x^2\cosh^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\cosh^{-1}\frac{x}{a} + \frac{(x^2 + 2a^2)\sqrt{x^2 - a^2}}{9}$ $[\cosh^{-1}(x/a) \lt 0]$

$\displaystyle \int \frac{\cosh^{-1}(x/a)}{x}\,dx = \frac{\ln^2(2x/a)}{2} + \frac{(a/x)^2}{2\cdot 2\cdot 2} + \frac{1\cdot 3\,(a/x)^4}{2\cdot 4\cdot 4\cdot 4} + \frac{1\cdot 3\cdot 5\,(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6} + \cdots$ $[\cosh^{-1}(x/a) > 0]$

$\displaystyle \int \frac{\cosh^{-1}(x/a)}{x}\,dx = -\left[\frac{\ln^2(2x/a)}{2} + \frac{(a/x)^2}{2\cdot 2\cdot 2} + \frac{1\cdot 3\,(a/x)^4}{2\cdot 4\cdot 4\cdot 4} + \frac{1\cdot 3\cdot 5\,(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6} + \cdots\right]$ $[\cosh^{-1}(x/a) \lt 0]$

$\displaystyle \int \frac{\cosh^{-1}(x/a)}{x^2}\,dx = -\frac{\cosh^{-1}(x/a)}{x} + \frac{1}{a}\sec^{-1}\frac{x}{a}$ $[\cosh^{-1}(x/a) > 0]$

$\displaystyle \int \frac{\cosh^{-1}(x/a)}{x^2}\,dx = -\frac{\cosh^{-1}(x/a)}{x} - \frac{1}{a}\sec^{-1}\frac{x}{a}$ $[\cosh^{-1}(x/a) \lt 0]$

$\displaystyle \int \tanh^{-1}\frac{x}{a}\,dx = x\tanh^{-1}\frac{x}{a} + \frac{a}{2}\ln(a^2 -\nobreak x^2)$

$\displaystyle \int x\tanh^{-1}\frac{x}{a}\,dx = \frac{ax}{2} + \frac{1}{2}(x^2 -\nobreak a^2)\tanh^{-1}\frac{x}{a}$

$\displaystyle \int x^2\tanh^{-1}\frac{x}{a}\,dx = \frac{ax^2}{6} + \frac{x^3}{3}\tanh^{-1}\frac{x}{a} + \frac{a^3}{6}\ln(a^2 -\nobreak x^2)$

$\displaystyle \int \frac{\tanh^{-1}(x/a)}{x}\,dx = \frac{x}{a} + \frac{(x/a)^3}{3^2} + \frac{(x/a)^5}{5^2} + \frac{(x/a)^7}{7^2} + \cdots$ $[|x| \lt a]$

$\displaystyle \int \frac{\tanh^{-1}(x/a)}{x^2}\,dx = -\frac{1}{x}\tanh^{-1}\frac{x}{a} + \frac{1}{2a}\ln\left(\frac{x^2}{a^2 - x^2}\right)$

$\displaystyle \int \coth^{-1}\frac{x}{a}\,dx = x\coth^{-1}\frac{x}{a} + \frac{a}{2}\ln(x^2 -\nobreak a^2)$

$\displaystyle \int x\coth^{-1}\frac{x}{a}\,dx = \frac{ax}{2} + \frac{1}{2}(x^2 -\nobreak a^2)\coth^{-1}\frac{x}{a}$

$\displaystyle \int x^2\coth^{-1}\frac{x}{a}\,dx = \frac{ax^2}{6} + \frac{x^3}{3}\coth^{-1}\frac{x}{a} + \frac{a^3}{6}\ln(x^2 -\nobreak a^2)$

$\displaystyle \int \frac{\coth^{-1}(x/a)}{x}\,dx = -\left(\frac{a}{x} + \frac{(a/x)^3}{3^2} + \frac{(a/x)^5}{5^2} + \cdots\right)$ $[|x| > a]$

$\displaystyle \int \frac{\coth^{-1}(x/a)}{x^2}\,dx = -\frac{1}{x}\coth^{-1}\frac{x}{a} + \frac{1}{2a}\ln\left(\frac{x^2}{x^2 - a^2}\right)$

$\displaystyle \int \operatorname{sech}^{-1}\frac{x}{a}\,dx = x\operatorname{sech}^{-1}\frac{x}{a} + a\sin^{-1}\frac{x}{a}$ $[\operatorname{sech}^{-1}(x/a) > 0]$

$\displaystyle \int \operatorname{sech}^{-1}\frac{x}{a}\,dx = x\operatorname{sech}^{-1}\frac{x}{a} - a\sin^{-1}\frac{x}{a}$ $[\operatorname{sech}^{-1}(x/a) \lt 0]$

$\displaystyle \int x\operatorname{sech}^{-1}\frac{x}{a}\,dx = \frac{x^2}{2}\operatorname{sech}^{-1}\frac{x}{a} - \frac{a\sqrt{a^2 - x^2}}{2}$ $[\operatorname{sech}^{-1}(x/a) > 0]$

$\displaystyle \int x\operatorname{sech}^{-1}\frac{x}{a}\,dx = \frac{x^2}{2}\operatorname{sech}^{-1}\frac{x}{a} + \frac{a\sqrt{a^2 - x^2}}{2}$ $[\operatorname{sech}^{-1}(x/a) \lt 0]$

$\displaystyle \int \frac{\operatorname{sech}^{-1}(x/a)}{x}\,dx = -\frac{1}{2}\ln\frac{a}{x}\,\ln\frac{4a}{x} - \frac{(x/a)^2}{2\cdot 2\cdot 2} - \frac{1\cdot 3\,(x/a)^4}{2\cdot 4\cdot 4\cdot 4} - \cdots$ $[\operatorname{sech}^{-1}(x/a) > 0]$

$\displaystyle \int \frac{\operatorname{sech}^{-1}(x/a)}{x}\,dx = \frac{1}{2}\ln\frac{a}{x}\,\ln\frac{4a}{x} + \frac{(x/a)^2}{2\cdot 2\cdot 2} + \frac{1\cdot 3\,(x/a)^4}{2\cdot 4\cdot 4\cdot 4} + \cdots$ $[\operatorname{sech}^{-1}(x/a) \lt 0]$

$\displaystyle \int \operatorname{csch}^{-1}\frac{x}{a}\,dx = x\operatorname{csch}^{-1}\frac{x}{a} \pm a\sinh^{-1}\frac{x}{a}$ [$+$ if $x > 0$, $-$ if $x \lt 0$]

$\displaystyle \int x\operatorname{csch}^{-1}\frac{x}{a}\,dx = \frac{x^2}{2}\operatorname{csch}^{-1}\frac{x}{a} \pm \frac{a\sqrt{x^2 + a^2}}{2}$ [$+$ if $x > 0$, $-$ if $x \lt 0$]

$\displaystyle \int \frac{\operatorname{csch}^{-1}(x/a)}{x}\,dx = \frac{1}{2}\ln\frac{x}{a}\,\ln\frac{4a}{x} + \frac{(x/a)^2}{2\cdot 2\cdot 2} - \frac{1\cdot 3\,(x/a)^4}{2\cdot 4\cdot 4\cdot 4} + \cdots$ $[0 \lt x \lt a]$

$\displaystyle \int \frac{\operatorname{csch}^{-1}(x/a)}{x}\,dx = \frac{1}{2}\ln\left(-\frac{x}{a}\right)\ln\left(-\frac{x}{4a}\right) - \frac{(x/a)^2}{2\cdot 2\cdot 2} + \frac{1\cdot 3\,(x/a)^4}{2\cdot 4\cdot 4\cdot 4} - \cdots$ $[-a \lt x \lt 0]$

$\displaystyle \int \frac{\operatorname{csch}^{-1}(x/a)}{x}\,dx = -\frac{a}{x} + \frac{(a/x)^3}{2\cdot 3\cdot 3} - \frac{1\cdot 3\,(a/x)^5}{2\cdot 4\cdot 5\cdot 5} + \cdots$ $[|x| > a]$

$\displaystyle \int x^m\sinh^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\sinh^{-1}\frac{x}{a} - \frac{1}{m + 1}\int \frac{x^{m+1}}{\sqrt{x^2 + a^2}}\,dx$

$\displaystyle \int x^m\cosh^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\cosh^{-1}\frac{x}{a} - \frac{1}{m + 1}\int \frac{x^{m+1}}{\sqrt{x^2 - a^2}}\,dx$ $[\cosh^{-1}(x/a) > 0]$

$\displaystyle \int x^m\cosh^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\cosh^{-1}\frac{x}{a} + \frac{1}{m + 1}\int \frac{x^{m+1}}{\sqrt{x^2 - a^2}}\,dx$ $[\cosh^{-1}(x/a) \lt 0]$

$\displaystyle \int x^m\tanh^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\tanh^{-1}\frac{x}{a} - \frac{a}{m + 1}\int \frac{x^{m+1}}{a^2 - x^2}\,dx$

$\displaystyle \int x^m\coth^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\coth^{-1}\frac{x}{a} - \frac{a}{m + 1}\int \frac{x^{m+1}}{a^2 - x^2}\,dx$

$\displaystyle \int x^m\operatorname{sech}^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\operatorname{sech}^{-1}\frac{x}{a} + \frac{a}{m + 1}\int \frac{x^m\,dx}{\sqrt{a^2 - x^2}}$ $[\operatorname{sech}^{-1}(x/a) > 0]$

$\displaystyle \int x^m\operatorname{sech}^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\operatorname{sech}^{-1}\frac{x}{a} - \frac{a}{m + 1}\int \frac{x^m\,dx}{\sqrt{a^2 - x^2}}$ $[\operatorname{sech}^{-1}(x/a) \lt 0]$

$\displaystyle \int x^m\operatorname{csch}^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\operatorname{csch}^{-1}\frac{x}{a} \pm \frac{a}{m + 1}\int \frac{x^m\,dx}{\sqrt{x^2 + a^2}}$ [$+$ if $x > 0$, $-$ if $x \lt 0$]