双曲線関数の原始関数の一覧
also: ∫ cosh n  a x d x = − 1 a ( n + 1 ) sinh  a x cosh n + 1  a x − n + 2 n + 1 ∫ cosh n + 2  a x d x (for n < 0 , n ≠ − 1 ) {\displaystyle \int \cosh ^{n}ax\,dx=-{\frac {1}{a(n+1)}}\sinh ax\cosh ^{n+1}ax-{\frac {n+2}{n+1}}\int \cosh ^{n+2}ax\,dx\qquad {\mbox{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}
∫ d x sinh  a x = 1 a ln  | tanh  a x 2 | + C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\,}
also: ∫ d x sinh  a x = 1 a ln  | cosh  a x − 1 sinh  a x | + C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+C\,}
∫ d x cosh n  a x = sinh  a x a ( n − 1 ) cosh n − 1  a x + n − 2 n − 1 ∫ d x cosh n − 2  a x (for n ≠ 1 ) {\displaystyle \int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
∫ cosh n  a x sinh m  a x d x = cosh n − 1  a x a ( n − m ) sinh m − 1  a x + n − 1 n − m ∫ cosh n − 2  a x sinh m  a x d x (for m ≠ n ) {\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,}
∫ sinh m  a x cosh n  a x d x = sinh m − 1  a x a ( m − n ) cosh n − 1  a x + m − 1 n − m ∫ sinh m − 2  a x cosh n  a x d x (for m ≠ n ) {\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,}
also: ∫ sinh m  a x cosh n  a x d x = sinh m + 1  a x a ( n − 1 ) cosh n − 1  a x + m − n + 2 n − 1 ∫ sinh m  a x cosh n − 2  a x d x (for n ≠ 1 ) {\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}