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