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