無理関数の原始関数の一覧
∫ S x d x = { 2 ( S − b a r c o t h ( S b ) ) (for b > 0 , a x > 0 ) 2 ( S − b a r t a n h ( S b ) ) (for b > 0 , a x < 0 ) 2 ( S − − b arctan  ( S − b ) ) (for b < 0 ) {\displaystyle \int {\frac {S}{x}}\,dx={\begin{cases}2\left(S-{\sqrt {b}}\,\mathrm {arcoth} \left({\frac {S}{\sqrt {b}}}\right)\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\2\left(S-{\sqrt {b}}\,\mathrm {artanh} \left({\frac {S}{\sqrt {b}}}\right)\right)&{\mbox{(for }}b>0,\quad ax<0{\mbox{)}}\\2\left(S-{\sqrt {-b}}\arctan \left({\frac {S}{\sqrt {-b}}}\right)\right)&{\mbox{(for }}b<0{\mbox{)}}\\\end{cases}}}
∫ x 4 d x s 3 = x s 2 − a 2 x s + 3 2 a 2 ln  | x + s a | {\displaystyle \int {\frac {x^{4}\;dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|}
∫ x 4 d x s 5 = − x s − 1 3 x 3 s 3 + ln  | x + s a | {\displaystyle \int {\frac {x^{4}\;dx}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|}
∫ x 2 m d x s 2 n + 1 = ( − 1 ) n − m 1 a 2 ( n − m ) ∑ i = 0 n − m − 1 1 2 ( m + i ) + 1 ( n − m − 1 i ) x 2 ( m + i ) + 1 s 2 ( m + i ) + 1 ( n > m ≥ 0 ) {\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad {\mbox{(}}n>m\geq 0{\mbox{)}}}
∫ d x u = arcsin  x a ( | x | ≤ | a | ) {\displaystyle \int {\frac {dx}{u}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫ x 2 d x u = 1 2 ( − x u + a 2 arcsin  x a ) ( | x | ≤ | a | ) {\displaystyle \int {\frac {x^{2}\;dx}{u}}={\frac {1}{2}}\left(-xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫ u d x = 1 2 ( x u − sgn  x arcosh  | x a | ) (for | x | ≥ | a | ) {\displaystyle \int u\;dx={\frac {1}{2}}\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(for }}|x|\geq |a|{\mbox{)}}}
∫ x u d x = − u ( | x | ≤ | a | ) {\displaystyle \int {\frac {x}{u}}\;dx=-u\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}