逆三角関数の原始関数の一覧
∫ x 2 arccsc  a x d x = x 3 arccsc  a x 3 + 1 6 a 3 arctanh 1 − 1 a 2 x 2 + x 2 6 a 1 − 1 a 2 x 2 + C {\displaystyle \int x^{2}\operatorname {arccsc} ax\,dx={\frac {x^{3}\operatorname {arccsc} ax}{3}}\,+\,{\frac {1}{6\,a^{3}}}\,\operatorname {arctanh} \,{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}\,+\,{\frac {x^{2}}{6\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}\,+\,C}
∫ x m arccsc  a x d x = x m + 1 arccsc  a x m + 1 + 1 a ( m + 1 ) ∫ x m − 1 1 − 1 a 2 x 2 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\operatorname {arccsc} ax\,dx={\frac {x^{m+1}\operatorname {arccsc} ax}{m+1}}\,+\,{\frac {1}{a\,(m+1)}}\int {\frac {x^{m-1}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}}\,dx\quad (m\neq -1)}