Collective Table of Formulas

Indefinite Integrals with hyperbolic cotangent (coth x)

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coth x
$ \int coth ax dx=\dfrac{\ln sh ax}{a} +C $
$ \int coth^{2} ax dx=x-\dfrac{coth ax}{a} +C $
$ \int coth^{3} ax dx=\dfrac{1}{a}\dfrac{\ln sh ax}{a}-\dfrac{coth^{2} ax}{2a} +C $
$ \int\dfrac{coth^{n} ax}{sh^{2} ax} dx=\dfrac{coth^{n+1} ax}{(n+1)a} +C $
$ \int\dfrac{dx}{coth ax sh^{2} ax} dx=\dfrac{1}{a}\ln coth ax +C $
$ \int\dfrac{dx}{coth ax} dx=\dfrac{1}{a}\ln ch ax +C $
$ \int x coth ax dx=\dfrac{1}{a^{2}}\biggl\{ ax+\dfrac{(ax)^{3}}{9}-\dfrac{(ax)^{5}}{225}+\dfrac{2(ax)^{7}}{105}+\cdots\dfrac{(-1)^{n-1}2^{2n}B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} +C $
$ \int x coth^{2} ax dx=\dfrac{x^{2}}{2}-\dfrac{x coth ax}{a}+\dfrac{1}{a^{2}}\ln sh ax+C $
$ \int\dfrac{coth ax}{x} dx=\biggl\{-\dfrac{1}{ax}+\dfrac{ax}{3}-\dfrac{(ax)^{3}}{135}+\cdots\dfrac{(-1)^{n-1}2^{2n}B_{n}(ax)^{2n-1}}{(2n-1)(2n)!}\biggl\} +C $
$ \int\dfrac{dx}{p+q coth ax}=\dfrac{px}{p^{2}-q^{2}}-\dfrac{q}{a(p^{2}-q^{2})}\ln(p sh ax+q ch ax) +C $
$ \int coth^{n} ax dx=-\dfrac{coth^{n-1} ax}{a(n-1)}+ \int coth^{n-2} ax dx $
Inverse Hyperbolic Cotangent ( arg coth x)
$ \int\arg coth\dfrac{x}{a}dx=x\arg coth x+\dfrac{a}{2}\ln(x^{2}-a^{2}) +C $
$ \int x\arg coth\dfrac{x}{a} dx=\dfrac{ax}{2}+\frac{1}{2}(x^{2}-a^{2})\arg coth\dfrac{x}{a} +C $
$ \int x^{2}\arg coth\dfrac{x}{a} dx=\dfrac{ax^{2}}{6}+\frac{a^{3}}{6}\ln(x^{2}-a^{2})+\dfrac{x^{3}}{3}\arg coth\dfrac{x}{a} +C $
$ \int\dfrac{\arg coth\dfrac{x}{a}}{x}dx=-\Biggl(\dfrac{a}{x}+\dfrac{(\dfrac{a}{x})^{3}}{3^{2}}+\dfrac{(\dfrac{a}{x})^{5}}{5^{2}}+\cdots\Biggl) +C $
$ \int\dfrac{\arg coth\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg coth\dfrac{x}{a}}{x}+\dfrac{1}{2a}\ln\Biggl(\dfrac{x^{2}}{x^{2}-a^{2}}\Biggl) +C $

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