逆三角関数の原始関数の一覧
∫ ( arcsin  a x ) n d x = x ( arcsin  a x ) n + n 1 − a 2 x 2 ( arcsin  a x ) n − 1 a − n ( n − 1 ) ∫ ( arcsin  a x ) n − 2 d x {\displaystyle \int (\arcsin ax)^{n}\,dx=x\,(\arcsin ax)^{n}\,+\,{\frac {n{\sqrt {1-a^{2}x^{2}}}\,(\arcsin ax)^{n-1}}{a}}\,-\,n\,(n-1)\int (\arcsin ax)^{n-2}\,dx}
∫ ( arcsin  a x ) n d x = x ( arcsin  a x ) n + 2 ( n + 1 ) ( n + 2 ) + 1 − a 2 x 2 ( arcsin  a x ) n + 1 a ( n + 1 ) − 1 ( n + 1 ) ( n + 2 ) ∫ ( arcsin  a x ) n + 2 d x ( n ≠ − 1 , − 2 ) {\displaystyle \int (\arcsin ax)^{n}\,dx={\frac {x\,(\arcsin ax)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {1-a^{2}x^{2}}}\,(\arcsin ax)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int (\arcsin ax)^{n+2}\,dx\quad (n\neq -1,-2)}
∫ arccsc  x d x = x arccsc  x + ln  | x + x 2 − 1 | + C = x arccsc  x + arccosh  ( x ) + C {\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x\,+\,\ln \left|x+{\sqrt {x^{2}-1}}\right|\,+\,C=x\operatorname {arccsc} x\,+\,\operatorname {arccosh} (x)\,+\,C}
∫ arccsc  a x d x = x arccsc  a x + 1 a arctanh 1 − 1 a 2 x 2 + C {\displaystyle \int \operatorname {arccsc} ax\,dx=x\operatorname {arccsc} ax+{\frac {1}{a}}\,\operatorname {arctanh} \,{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}
∫ x arccsc  a x d x = x 2 arccsc  a x 2 + x 2 a 1 − 1 a 2 x 2 + C {\displaystyle \int x\operatorname {arccsc} ax\,dx={\frac {x^{2}\operatorname {arccsc} ax}{2}}+{\frac {x}{2\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}