有理関数の原始関数の一覧
∫ x a x 2 + b x + c d x = 1 2 a ln  | a x 2 + b x + c | − b 2 a ∫ d x a x 2 + b x + c + C {\displaystyle \int {\frac {x}{ax^{2}+bx+c}}dx={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {dx}{ax^{2}+bx+c}}+C}
∫ m x + n a x 2 + b x + c d x = { m 2 a ln  | a x 2 + b x + c | + 2 a n − b m a 4 a c − b 2 arctan  2 a x + b 4 a c − b 2 + C (for 4 a c − b 2 > 0 ) m 2 a ln  | a x 2 + b x + c | − 2 a n − b m a b 2 − 4 a c a r c t a n h 2 a x + b b 2 − 4 a c + C (for 4 a c − b 2 < 0 ) m 2 a ln  | a x 2 + b x + c | − 2 a n − b m a ( 2 a x + b ) + C (for 4 a c − b 2 = 0 ) {\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\begin{cases}{\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {4ac-b^{2}}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}+C&{\mbox{(for }}4ac-b^{2}>0{\mbox{)}}\\{\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+C&{\mbox{(for }}4ac-b^{2}<0{\mbox{)}}\\{\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}+C&{\mbox{(for }}4ac-b^{2}=0{\mbox{)}}\end{cases}}}