無理関数の原始関数の一覧
( x 2 > a 2 ) {\displaystyle (x^{2}>a^{2})} を前提とする。 ( x 2 < a 2 ) {\displaystyle (x^{2}<a^{2})} の場合は次節:
∫ s d x = 1 2 ( x s − a 2 ln  ( x + s ) ) {\displaystyle \int s\;dx={\frac {1}{2}}\left(xs-a^{2}\ln(x+s)\right)}
∫ x s d x = 1 3 s 3 {\displaystyle \int xs\;dx={\frac {1}{3}}s^{3}}
∫ s d x x = s − a arccos  | a x | {\displaystyle \int {\frac {s\;dx}{x}}=s-a\arccos \left|{\frac {a}{x}}\right|}
∫ d x s = ln  | x + s a | {\displaystyle \int {\frac {dx}{s}}=\ln \left|{\frac {x+s}{a}}\right|}
ここで、 ln  | x + s a | = s g n ( x ) arcosh  | x a | = 1 2 ln  ( x + s x − s ) {\displaystyle \ln \left|{\frac {x+s}{a}}\right|=\mathrm {sgn} (x)\,\operatorname {arcosh} \left|{\frac {x}{a}}\right|={\frac {1}{2}}\ln \left({\frac {x+s}{x-s}}\right)} , arcosh  | x a | {\displaystyle \operatorname {arcosh} \left|{\frac {x}{a}}\right|} の正の値を採用する。
∫ x 5 r d x = r 7 7 − 2 a 2 r 5 5 + a 4 r 3 3 {\displaystyle \int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}
∫ x 5 r 3 d x = r 9 9 − 2 a 2 r 7 7 + a 4 r 5 5 {\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}
∫ x 5 r 2 n + 1 d x = r 2 n + 7 2 n + 7 − 2 a 2 r 2 n + 5 2 n + 5 + a 4 r 2 n + 3 2 n + 3 {\displaystyle \int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}
∫ r d x x = r − a ln  | a + r x | = r − a arsinh  a x {\displaystyle \int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\,\operatorname {arsinh} {\frac {a}{x}}}
∫ x d x s = s {\displaystyle \int {\frac {x\;dx}{s}}=s}
∫ x d x s 3 = − 1 s {\displaystyle \int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}}
∫ x d x s 5 = − 1 3 s 3 {\displaystyle \int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}
∫ x d x s 7 = − 1 5 s 5 {\displaystyle \int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}
∫ x d x s 2 n + 1 = − 1 ( 2 n − 1 ) s 2 n − 1 {\displaystyle \int {\frac {x\;dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}