無理関数の原始関数の一覧
∫ S d x = 2 S 3 3 a {\displaystyle \int S{dx}={\frac {2S^{3}}{3a}}}
∫ d x S = 2 S a {\displaystyle \int {\frac {dx}{S}}={\frac {2S}{a}}}
∫ d x x S = { − 2 b a r c o t h ( S b ) (for b > 0 , a x > 0 ) − 2 b a r t a n h ( S b ) (for b > 0 , a x < 0 ) 2 − b arctan  ( S − b ) (for b < 0 ) {\displaystyle \int {\frac {dx}{xS}}={\begin{cases}-{\frac {2}{\sqrt {b}}}\mathrm {arcoth} \left({\frac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\-{\frac {2}{\sqrt {b}}}\mathrm {artanh} \left({\frac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax<0{\mbox{)}}\\{\frac {2}{\sqrt {-b}}}\arctan \left({\frac {S}{\sqrt {-b}}}\right)&{\mbox{(for }}b<0{\mbox{)}}\\\end{cases}}}