反三角函数积分表 from 维基

\int \arcsin \frac{x}{c} \  dx = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2}

\int x \arcsin \frac{x}{c} \  dx = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arcsin \frac{x}{c} + \frac{x}{4} \sqrt{c^2 - x^2}

\int x^2 \arcsin \frac{x}{c} \  dx = \frac{x^3}{3} \arcsin \frac{x}{c} + \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}

\int x^n \arcsin x \  dx = \frac{1}{n + 1} \left( x^{n + 1} \arcsin x + \frac{x^n \sqrt{1 - x^2} - n x^{n - 1} \arcsin x}{n - 1} + n \int x^{n - 2} \arcsin x \  dx \right)

\int \arccos \frac{x}{c} \  dx = x \arccos \frac{x}{c} - \sqrt{c^2 - x^2}

\int x \arccos \frac{x}{c} \  dx = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arccos \frac{x}{c} - \frac{x}{4} \sqrt{c^2 - x^2}

\int x^2 \arccos \frac{x}{c} \  dx = \frac{x^3}{3} \arccos \frac{x}{c} - \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}

\int \arctan \frac{x}{c} \  dx = x \arctan \frac{x}{c} - \frac{c}{2} \ln(c^2 + x^2)

\int x \arctan \frac{x}{c} \  dx = \frac{c^2 + x^2}{2} \arctan \frac{x}{c} - \frac{c x}{2}

\int x^2 \arctan \frac{x}{c} \  dx = \frac{x^3}{3} \arctan \frac{x}{c} - \frac{c x^2}{6} + \frac{c^3}{6} \ln{c^2 + x^2}

\int x^n \arctan \frac{x}{c} \  dx = \frac{x^{n + 1}}{n + 1} \arctan \frac{x}{c} - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \  dx, \quad n \neq 1

\int \arcsec \frac{x}{c} \  dx = x \arcsec \frac{x}{c} + \frac{x}{c |x|} \ln \left| x \pm \sqrt{x^2 - 1} \right|

\int x \arcsec x \  dx = \frac{1}{2} \left( x^2 \arcsec x - \sqrt{x^2 - 1} \right)

\int \arccot \frac{x}{c} \  dx = x \arccot \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2)

\int x \arccot \frac{x}{c} \  dx = \frac{c^2 + x^2}{2} \arccot \frac{x}{c} + \frac{c x}{2}

\int x^2 \arccot \frac{x}{c} \  dx = \frac{x^3}{3} \arccot \frac{x}{c} + \frac{c x^2}{6} - \frac{c^3}{6} \ln(c^2 + x^2)

\int x^n \arccot \frac{x}{c} \  dx = \frac{x^{n + 1}}{n+1} \arccot \frac{x}{c} + \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \  dx, \quad n \neq 1