List of integrals of arc hyperbolic functions

The following is a list of integrals (antiderivative functions) of inverse hyperbolic functions. For a complete list of integral functions, see lists of integrals.

: $intmathrm{arsinh},frac{x}{c},dx = x,mathrm{arsinh},frac{x}{c}-sqrt{x^2+c^2}$

: $intmathrm{arcosh},frac{x}{c},dx = x,mathrm{arcosh},frac{x}{c}-sqrt{x^2-c^2}$

: $intmathrm{artanh},frac{x}{c},dx = x,mathrm{artanh},frac{x}{c} + frac{c}{2}ln|c^2 - x^2| qquadmbox{(for }|x|<|c|mbox{)}$

: $intmathrm{arcoth},frac{x}{c},dx = x,mathrm{arcoth},frac{x}{c} + frac{c}{2}ln|x^2 - c^2| qquadmbox{(for }|x|>|c|mbox{)}$

: $intmathrm{arsech},frac{x}{c},dx = x,mathrm{arsech},frac{x}{c} - c,mathrm{arctan},frac{x,sqrt{frac{c - x}{c + x}{x - c} qquadmbox{(for } x in (0,,c) mbox{)}$

: $intmathrm{arcsch},frac{x}{c},dx = x,mathrm{arcsch},frac{x}{c} + c,ln,frac{x + sqrt{x^2 + c^2{c} qquadmbox{(for } x in (0,,c) mbox{)}$

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