List of integrals of hyperbolic functions

The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals.

In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.

Integrals involving only hyperbolic sine functions

Source:

• $\int \sinh ax\,dx={\frac {1}{a}}\cosh ax+C$
• $\int \sinh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+C$
• $\int \sinh ^{n}ax\,dx={\begin{cases}{\frac {1}{an}}(\sinh ^{n-1}ax)(\cosh ax)-{\frac {n-1}{n}}\displaystyle \int \sinh ^{n-2}ax\,dx,&n>0\\{\frac {1}{a(n+1)}}(\sinh ^{n+1}ax)(\cosh ax)-{\frac {n+2}{n+1}}\displaystyle \int \sinh ^{n+2}ax\,dx,&n<0,n\neq -1\end{cases}}$
• ${\begin{aligned}\int {\frac {dx}{\sinh ax}}&={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\\&={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+C\\&={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\\&={\frac {1}{2a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\end{aligned}}$
• $\int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}$
• $\int x\sinh ax\,dx={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+C$
• $\int (\sinh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh bx)(\cosh ax)-b(\cosh bx)(\sinh ax){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}$

Integrals involving only hyperbolic cosine functions

Source:

• $\int \cosh ax\,dx={\frac {1}{a}}\sinh ax+C$
• $\int \cosh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+C$
• $\int \cosh ^{n}ax\,dx={\begin{cases}{\frac {1}{an}}(\sinh ax)(\cosh ^{n-1}ax)+{\frac {n-1}{n}}\displaystyle \int \cosh ^{n-2}ax\,dx,&n>0\\-{\frac {1}{a(n+1)}}(\sinh ax)(\cosh ^{n+1}ax)+{\frac {n+2}{n+1}}\displaystyle \int \cosh ^{n+2}ax\,dx,&n<0,n\neq -1\end{cases}}$
• ${\begin{aligned}\int {\frac {dx}{\cosh ax}}&={\frac {2}{a}}\arctan e^{ax}+C\\&={\frac {1}{a}}\arctan(\sinh ax)+C\end{aligned}}$
• $\int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}$
• $\int x\cosh ax\,dx={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+C$
• $\int x^{2}\cosh ax\,dx=-{\frac {2x\cosh ax}{a^{2}}}+\left({\frac {x^{2}}{a}}+{\frac {2}{a^{3}}}\right)\sinh ax+C$
• $\int (\cosh ax)(\cosh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax)(\cosh bx)-b(\sinh bx)(\cosh ax){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}$
• $\int {\frac {dx}{1+\cosh(ax)}}={\frac {2}{a}}{\frac {1}{1+e^{-ax}}}+C\quad$ or ${\frac {2}{a}}$ times The Logistic Function

Other integrals

Integrals of hyperbolic tangent, cotangent, secant, cosecant functions

Source:

• $\int \tanh x\,dx=\ln \cosh x+C$
• $\int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a}}+C$
• $\int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}$
• $\int \coth x\,dx=\ln |\sinh x|+C,{\text{ for }}x\neq 0$
• $\int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}$
• $\int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C$
• $\int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C=\ln \left|\coth {x}-\operatorname {csch} {x}\right|+C,{\text{ for }}x\neq 0$

Integrals involving hyperbolic sine and cosine functions

Source:

• $\int (\cosh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax)(\sinh bx)-b(\cosh ax)(\cosh bx){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}$
• ${\begin{aligned}\int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}\,dx&={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax}}\,dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\\&=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax}}\,dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\\&=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax}}\,dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\end{aligned}}$
• ${\begin{aligned}\int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}\,dx&={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}\,dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\\&={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\\&=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax}}\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\end{aligned}}$

Integrals involving hyperbolic and trigonometric functions

• $\int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+C$
• $\int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)+C$
• $\int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+C$
• $\int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)+C$