list of integrals of area functions

The following is a list of integrals (antiderivative functions) of area functions. For a complete list of integral functions, please see table of integrals and list of integrals.

\int\mathrm{arsinh}\,\frac{x}{c}\,dx = x\,\mathrm{arsinh}\,\frac{x}{c}-\sqrt{x^2+c^2}

\int\mathrm{arcosh}\,\frac{x}{c}\,dx = x\,\mathrm{arcosh}\,\frac{x}{c}-\sqrt{x^2-c^2}

\int\mathrm{artanh}\,\frac{x}{c}\,dx = x\,\mathrm{artanh}\,\frac{x}{c} + \frac{c}{2}\ln|c^2 - x^2| \qquad\mbox{(for }|x|<|c|\mbox{)}

\int\mathrm{arcoth}\,\frac{x}{c}\,dx = x\,\mathrm{arcoth}\,\frac{x}{c} + \frac{c}{2}\ln|x^2 - c^2| \qquad\mbox{(for }|x|>|c|\mbox{)}

\int\mathrm{arsech}\,\frac{x}{c}\,dx = x\,\mathrm{arsech}\,\frac{x}{c} - c\,\mathrm{arctan}\,\frac{x\,\sqrt{\frac{c - x}{c + x}}}{x - c} \qquad\mbox{(for } x \in (0,\,c) \mbox{)}

\int\mathrm{arcsch}\,\frac{x}{c}\,dx = x\,\mathrm{arcsch}\,\frac{x}{c} + c\,\ln\,\frac{x + \sqrt{x^2 + c^2}}{c} \qquad\mbox{(for } x \in (0,\,c) \mbox{)}

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