双曲線関数の原始関数の一覧
also: ∫ d x sinh  a x = 1 a ln  | sinh  a x cosh  a x + 1 | + C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\,}
also: ∫ d x sinh  a x = 1 a ln  | cosh  a x − 1 cosh  a x + 1 | + C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\,}
∫ d x cosh  a x = 2 a arctan  e a x + C = 1 a gd  ( a x ) + C gd  x {\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+C={\frac {1}{a}}\operatorname {gd} (ax)+C\quad \operatorname {gd} x} :グーデルマン関数
∫ d x sinh n  a x = − cosh  a x a ( n − 1 ) sinh n − 1  a x − n − 2 n − 1 ∫ d x sinh n − 2  a x (for n ≠ 1 ) {\displaystyle \int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
∫ coth n  a x d x = − 1 a ( n − 1 ) coth n − 1  a x + ∫ coth n − 2  a x d x (for n ≠ 1 ) {\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
∫ sinh  a x sinh  b x d x = 1 a 2 − b 2 ( a sinh  b x cosh  a x − b cosh  b x sinh  a x ) + C (for a 2 ≠ b 2 ) {\displaystyle \int \sinh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh bx\cosh ax-b\cosh bx\sinh ax)+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}\,}
∫ cosh  a x cosh  b x d x = 1 a 2 − b 2 ( a sinh  a x cosh  b x − b sinh  b x cosh  a x ) + C (for a 2 ≠ b 2 ) {\displaystyle \int \cosh ax\cosh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\cosh bx-b\sinh bx\cosh ax)+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}\,}