List of integrals of inverse hyperbolic functions

The following is a list of integrals (antiderivative 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.

$\int \sinh axdx=\frac{1}{a}\cosh ax+C$

$\int \cosh axdx=\frac{1}{a}\sinh ax+C$

$\int {\sinh }^{2}axdx=\frac{1}{4a}\sinh 2ax-\frac{x}{2}+C$

$\int {\cosh }^{2}axdx=\frac{1}{4a}\sinh 2ax+\frac{x}{2}+C$

$\int {\tanh }^{2}axdx=x-\frac{\tanh ax}{a}+C$

$\int {\sinh }^{n}axdx=\frac{1}{an}{\sinh }^{n-1}ax\cosh ax-\frac{n-1}{n}\int {\sinh }^{}axdx\text{(for }n>0\text{)}$

also: $\int {\sinh }^{n}axdx=\frac{1}{a(n+1)}{\sinh }^{n+1}ax\cosh ax-\frac{n+2}{n+1}\int {\sinh }^{}axdx\text{(for }n<0\text{, }n\ne -1\text{)}$

$\int {\cosh }^{n}axdx=\frac{1}{an}\sinh ax{\cosh }^{}ax+\frac{n-1}{n}\int {\cosh }^{}axdx\text{(for }n>0\text{)}$

also: $\int {\cosh }^{n}axdx=-\frac{1}{a(n+1)}\sinh ax{\cosh }^{}ax+\frac{n+2}{n+1}\int {\cosh }^{}axdx\text{(for }n<0\text{, }n\ne -1\text{)}$

$\int \frac{dx}{\sinh ax}=\frac{1}{a}\ln |\tanh \frac{ax}{2}|+C$

also: $\int \frac{dx}{\sinh ax}=\frac{1}{a}\ln \left|\frac{\cosh ax-1}{\sinh ax}\right|+C$

also: $\int \frac{dx}{\sinh ax}=\frac{1}{a}\ln \left|\frac{\sinh ax}{\cosh ax+1}\right|+C$

also: $\int \frac{dx}{\sinh ax}=\frac{1}{2a}\ln \left|\frac{\cosh ax-1}{\cosh ax+1}\right|+C$

$\int \frac{dx}{\cosh ax}=\frac{2}{a}\arctan e^{ax}+C$

also: $\int \frac{dx}{\cosh ax}=\frac{1}{a}\arctan (\sinh ax)+C$

$\int \frac{dx}{{\sinh }^{n}ax}=-\frac{\cosh ax}{a(n-1){\sinh }^{n-1}ax}-\frac{n-2}{n-1}\int \frac{dx}{{\sinh }^{n-2}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{{\cosh }^{n}ax}=\frac{\sinh ax}{a(n-1){\cosh }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\cosh }^{n-2}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\cosh }^{n}ax}{{\sinh }^{m}ax}dx=\frac{{\cosh }^{n-1}ax}{a(n-m){\sinh }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\cosh }^{n-2}ax}{{\sinh }^{m}ax}dx\text{(for }m\ne n\text{)}$

also: $\int \frac{{\cosh }^{n}ax}{{\sinh }^{m}ax}dx=-\frac{{\cosh }^{n+1}ax}{a(m-1){\sinh }^{m-1}ax}+\frac{n-m+2}{m-1}\int \frac{{\cosh }^{n}ax}{{\sinh }^{m-2}ax}dx\text{(for }m\ne 1\text{)}$

also: $\int \frac{{\cosh }^{n}ax}{{\sinh }^{m}ax}dx=-\frac{{\cosh }^{n-1}ax}{a(m-1){\sinh }^{m-1}ax}+\frac{n-1}{m-1}\int \frac{{\cosh }^{n-2}ax}{{\sinh }^{m-2}ax}dx\text{(for }m\ne 1\text{)}$

$\int \frac{{\sinh }^{m}ax}{{\cosh }^{n}ax}dx=\frac{{\sinh }^{m-1}ax}{a(m-n){\cosh }^{n-1}ax}+\frac{m-1}{n-m}\int \frac{{\sinh }^{m-2}ax}{{\cosh }^{n}ax}dx\text{(for }m\ne n\text{)}$

also: $\int \frac{{\sinh }^{m}ax}{{\cosh }^{n}ax}dx=\frac{{\sinh }^{m+1}ax}{a(n-1){\cosh }^{n-1}ax}+\frac{m-n+2}{n-1}\int \frac{{\sinh }^{m}ax}{{\cosh }^{n-2}ax}dx\text{(for }n\ne 1\text{)}$

also: $\int \frac{{\sinh }^{m}ax}{{\cosh }^{n}ax}dx=-\frac{{\sinh }^{m-1}ax}{a(n-1){\cosh }^{n-1}ax}+\frac{m-1}{n-1}\int \frac{{\sinh }^{m-2}ax}{{\cosh }^{n-2}ax}dx\text{(for }n\ne 1\text{)}$

$\int x\sinh axdx=\frac{1}{a}x\cosh ax-\frac{1}{a^2}\sinh ax+C$

$\int x\cosh axdx=\frac{1}{a}x\sinh ax-\frac{1}{a^2}\cosh ax+C$

$\int x^2\cosh axdx=-\frac{2x\cosh ax}{a^2}+(\frac{x^2}{a}+\frac{2}{a^3})\sinh ax+C$

$\int \tanh axdx=\frac{1}{a}\ln \cosh ax+C$

$\int \coth axdx=\frac{1}{a}\ln |\sinh ax|+C$

$\int {\tanh }^{n}axdx=-\frac{1}{a(n-1)}{\tanh }^{n-1}ax+\int {\tanh }^{n-2}axdx\text{(for }n\ne 1\text{)}$

$\int {\coth }^{n}axdx=-\frac{1}{a(n-1)}{\coth }^{n-1}ax+\int {\coth }^{n-2}axdx\text{(for }n\ne 1\text{)}$

$\int \sinh ax\sinh bxdx=\frac{1}{a^2-b^2}(a\sinh bx\cosh ax-b\cosh bx\sinh ax)+C\text{(for }{a}^2\ne b^2\text{)}$

$\int \cosh ax\cosh bxdx=\frac{1}{a^2-b^2}(a\sinh ax\cosh bx-b\sinh bx\cosh ax)+C\text{(for }{a}^2\ne b^2\text{)}$

$\int \cosh ax\sinh bxdx=\frac{1}{a^2-b^2}(a\sinh ax\sinh bx-b\cosh ax\cosh bx)+C\text{(for }{a}^2\ne b^2\text{)}$

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

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

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

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

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