Annex: Integrals of hyperbolic inverse functions

The following is a list of integrals of the inverses of hyperbolic functions:

∫ ∫ arsinh xcdx=xarsinh xc− − x2+c2{displaystyle int operatorname {arsinh} {frac {x}{c}}{,dx=xoperatorname {arsinh} {frac {x}{x}{x}}{x^{2}+c^{2}}}}}}}}}}

∫ ∫ archsh xcdx=xarchsh xc− − x2− − c2{displaystyle int operatorname {arcosh} {frac {x}{c}}{,dx=xoperatorname {arcosh} {frac {x}{x}{x}{x^{2}-c^{2}}}}}}}}}}

<math alttext="{displaystyle int operatorname {artanh} {frac {x}{c}},dx=xoperatorname {artanh} {frac {x}{c}}+{frac {c}{2}}ln left|c^{2}-x^{2}right|qquad ({text{para }}|x|∫ ∫ artanh xcdx=xartanh xc+c2ln 日本語c2− − x2日本語(for日本語x日本語.日本語c日本語){displaystyle int operatorname {artanh} {frac {x}{c}{c}}},dx=xoperatorname {artanh} {frac {x}{x}{c}}}} +{frac {c}{2}}}}ln lephtac^{2}-x^{2}{right.<img alt="{displaystyle int operatorname {artanh} {frac {x}{c}},dx=xoperatorname {artanh} {frac {x}{c}}+{frac {c}{2}}ln left|c^{2}-x^{2}right|qquad ({text{para }}|x|

|c|)}" xmlns="http://www.w3.org/1998/Math/MathML">∫ ∫ archth xcdx=xarchth xc+c2ln 日本語x2− − c2日本語(for日本語x日本語▪日本語c日本語){displaystyle int operatorname {arcoth} {frac {x}{c}{c}}},dx=xoperatorname {arcoth} {frac {x}{x}{c}}}} +{frac {c}{2}}}ln lephtax^{2}-c^{2}{rightqquad ({ctext{cn}{cs)}{c(s)}{c(s)}}{c(s)}}{cs)}}{cs)}|c|)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc521268e3d8547191d58b393545e5240997354" style="vertical-align: -2.338ex; width:63.698ex; height:5.676ex;"/>

∫ ∫ arsech xcdx=xarsech xc− − carctan xc− − xc+xx− − c(forx한 한 (0,c)){displaystyle int operatorname {arsech} {frac}{x}{x}}},dx=xoperatorname {sech} {frac {x}{x}{c}{xoperatorname {arctan} {frac {x}{xx}{cd}{csq

∫ ∫ arcsch xcdx=xarcsch xc+clnx+x2+c2c(forx한 한 (0,c)){displaystyle int operatorname {arcsch} {frac {x}{c}}{,dx=xoperatorname {arcsch} {frac {x}{x}{x}}}{x}{cn}{cd}{x+{sqrt {x^{2}{2}{2}{x}{x}{x}{x