有理関数の原始関数の一覧
∫ x 2 ( a x + b ) 2 d x = 1 a 3 ( a x − 2 b ln  | a x + b | − b 2 a x + b ) + C {\displaystyle \int {\frac {x^{2}}{(ax+b)^{2}}}dx={\frac {1}{a^{3}}}\left(ax-2b\ln \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)+C}
∫ x 2 ( a x + b ) 3 d x = 1 a 3 ( ln  | a x + b | + 2 b a x + b − b 2 2 ( a x + b ) 2 ) + C {\displaystyle \int {\frac {x^{2}}{(ax+b)^{3}}}dx={\frac {1}{a^{3}}}\left(\ln \left|ax+b\right|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right)+C}
∫ x 2 ( a x + b ) n d x = 1 a 3 ( − ( a x + b ) 3 − n ( n − 3 ) + 2 b ( a x + b ) 2 − n ( n − 2 ) − b 2 ( a x + b ) 1 − n ( n − 1 ) ) + C (for n ∉ { 1 , 2 , 3 } ) {\displaystyle \int {\frac {x^{2}}{(ax+b)^{n}}}dx={\frac {1}{a^{3}}}\left(-{\frac {(ax+b)^{3-n}}{(n-3)}}+{\frac {2b(ax+b)^{2-n}}{(n-2)}}-{\frac {b^{2}(ax+b)^{1-n}}{(n-1)}}\right)+C\qquad {\mbox{(for }}n\not \in \{1,2,3\}{\mbox{)}}}
∫ 1 x 2 + a 2 d x = 1 a arctan  x a + C {\displaystyle \int {\frac {1}{x^{2}+a^{2}}}dx={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!+C}
∫ 1 x 2 − a 2 d x = { − 1 a a r c t a n h x a = 1 2 a ln  a − x a + x + C (for | x | < | a | ) − 1 a a r c c o t h x a = 1 2 a ln  x − a x + a + C (for | x | > | a | ) {\displaystyle \int {\frac {1}{x^{2}-a^{2}}}dx={\begin{cases}-{\frac {1}{a}}\,\mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}+C&{\mbox{(for }}|x|<|a|{\mbox{)}}\\-{\frac {1}{a}}\,\mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}+C&{\mbox{(for }}|x|>|a|{\mbox{)}}\end{cases}}}
For a ≠ 0 : {\displaystyle a\neq 0:}