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