List of integrals of hyperbolic functions

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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

• $\int \sinh axdx=\frac{1}{a}\cosh ax+C$
• $\int {\sinh }^{2}axdx=\frac{1}{4a}\sinh 2ax-\frac{x}{2}+C$
• $\int {\sinh }^{n}axdx=\{\begin{array}{ll}\frac{1}{an}({\sinh }^{n-1}ax)(\cosh ax)-\frac{n-1}{n}\int {\sinh }^{n-2}axdx,& n>0\\ \frac{1}{a(n+1)}({\sinh }^{n+1}ax)(\cosh ax)-\frac{n+2}{n+1}\int {\sinh }^{n+2}axdx,& n<0,n\ne -1\end{array}$
• $\begin{array}{rl}\int \frac{dx}{\sinh ax}& =\frac{1}{a}\ln |\tanh \frac{ax}{2}|+C\\ & =\frac{1}{a}\ln \left|\frac{\cosh ax-1}{\sinh ax}\right|+C\\ & =\frac{1}{a}\ln \left|\frac{\sinh ax}{\cosh ax+1}\right|+C\\ & =\frac{1}{2a}\ln \left|\frac{\cosh ax-1}{\cosh ax+1}\right|+C\end{array}$
• $\int \frac{dx}{{\sinh }^{n}ax}=-\frac{\cosh ax}{a(n-1){\sinh }^{n-1}ax}-\frac{n-2}{n-1}\int \frac{dx}{{\sinh }^{n-2}ax}\text{(for }n\ne 1\text{)}$
• $\int x\sinh axdx=\frac{1}{a}x\cosh ax-\frac{1}{a^2}\sinh ax+C$
• $\int (\sinh ax)(\sinh bx)dx=\frac{1}{a^2-b^2}(a(\sinh bx)(\cosh ax)-b(\cosh bx)(\sinh ax))+C\text{(for }{a}^2\ne b^2\text{)}$

Integrals involving only hyperbolic cosine functions

• $\int \cosh axdx=\frac{1}{a}\sinh ax+C$
• $\int {\cosh }^{2}axdx=\frac{1}{4a}\sinh 2ax+\frac{x}{2}+C$
• $\int {\cosh }^{n}axdx=\{\begin{array}{ll}\frac{1}{an}(\sinh ax)({\cosh }^{n-1}ax)+\frac{n-1}{n}\int {\cosh }^{n-2}axdx,& n>0\\ -\frac{1}{a(n+1)}(\sinh ax)({\cosh }^{n+1}ax)+\frac{n+2}{n+1}\int {\cosh }^{n+2}axdx,& n<0,n\ne -1\end{array}$
• $\begin{array}{rl}\int \frac{dx}{\cosh ax}& =\frac{2}{a}\arctan e^{ax}+C\\ & =\frac{1}{a}\arctan (\sinh ax)+C\end{array}$
• $\int \frac{dx}{{\cosh }^{n}ax}=\frac{\sinh ax}{a(n-1){\cosh }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\cosh }^{n-2}ax}\text{(for }n\ne 1\text{)}$
• $\int x\cosh axdx=\frac{1}{a}x\sinh ax-\frac{1}{a^2}\cosh ax+C$
• $\int x^2\cosh axdx=-\frac{2x\cosh ax}{a^2}+(\frac{x^2}{a}+\frac{2}{a^3})\sinh ax+C$
• $\int (\cosh ax)(\cosh bx)dx=\frac{1}{a^2-b^2}(a(\sinh ax)(\cosh bx)-b(\sinh bx)(\cosh ax))+C\text{(for }{a}^2\ne b^2\text{)}$
• $\int \frac{dx}{1+\cosh (ax)}=\frac{2}{a}\frac{1}{1+e^{-ax}}+C$ or $\frac{2}{a}$ times The Logistic Function

Other integrals

Integrals of hyperbolic tangent, cotangent, secant, cosecant functions

• $\int \tanh xdx=\ln \cosh x+C$
• $\int {\tanh }^{2}axdx=x-\frac{\tanh ax}{a}+C$
• $\int {\tanh }^{n}axdx=-\frac{1}{a(n-1)}{\tanh }^{n-1}ax+\int {\tanh }^{n-2}axdx\text{(for }n\ne 1\text{)}$
• $\int \coth xdx=\ln |\sinh x|+C,\text{ for }x\ne 0$
• $\int {\coth }^{n}axdx=-\frac{1}{a(n-1)}{\coth }^{n-1}ax+\int {\coth }^{n-2}axdx\text{(for }n\ne 1\text{)}$
• $\int \mathrm{sech}xdx=\arctan (\sinh x)+C$
• $\int \mathrm{csch}xdx=\ln |\tanh \frac{x}{2}|+C=\ln |\coth x-\mathrm{csch}x|+C,\text{ for }x\ne 0$

Integrals involving hyperbolic sine and cosine functions

• $\int (\cosh ax)(\sinh bx)dx=\frac{1}{a^2-b^2}(a(\sinh ax)(\sinh bx)-b(\cosh ax)(\cosh bx))+C\text{(for }{a}^2\ne b^2\text{)}$
• $\begin{array}{rl}\int \frac{{\cosh }^{n}ax}{{\sinh }^{m}ax}dx& =\frac{{\cosh }^{n-1}ax}{a(n-m){\sinh }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\cosh }^{n-2}ax}{{\sinh }^{m}ax}dx\text{(for }m\ne n\text{)}\\ & =-\frac{{\cosh }^{n+1}ax}{a(m-1){\sinh }^{m-1}ax}+\frac{n-m+2}{m-1}\int \frac{{\cosh }^{n}ax}{{\sinh }^{m-2}ax}dx\text{(for }m\ne 1\text{)}\\ & =-\frac{{\cosh }^{n-1}ax}{a(m-1){\sinh }^{m-1}ax}+\frac{n-1}{m-1}\int \frac{{\cosh }^{n-2}ax}{{\sinh }^{m-2}ax}dx\text{(for }m\ne 1\text{)}\end{array}$
• $\begin{array}{rl}\int \frac{{\sinh }^{m}ax}{{\cosh }^{n}ax}dx& =\frac{{\sinh }^{m-1}ax}{a(m-n){\cosh }^{n-1}ax}+\frac{m-1}{n-m}\int \frac{{\sinh }^{m-2}ax}{{\cosh }^{n}ax}dx\text{(for }m\ne n\text{)}\\ & =\frac{{\sinh }^{m+1}ax}{a(n-1){\cosh }^{n-1}ax}+\frac{m-n+2}{n-1}\int \frac{{\sinh }^{m}ax}{{\cosh }^{n-2}ax}dx\text{(for }n\ne 1\text{)}\\ & =-\frac{{\sinh }^{m-1}ax}{a(n-1){\cosh }^{n-1}ax}+\frac{m-1}{n-1}\int \frac{{\sinh }^{m-2}ax}{{\cosh }^{n-2}ax}dx\text{(for }n\ne 1\text{)}\end{array}$

Integrals involving hyperbolic and trigonometric functions

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

References