# List of integrals of inverse hyperbolic functions

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The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals.

## Contents

• In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
• For each inverse hyperbolic integration formula below there is a corresponding formula in the list of integrals of inverse trigonometric functions.
• ## Inverse hyperbolic sine integration formulas

arsinh ( a x ) d x = x arsinh ( a x ) a 2 x 2 + 1 a + C x arsinh ( a x ) d x = x 2 arsinh ( a x ) 2 + arsinh ( a x ) 4 a 2 x a 2 x 2 + 1 4 a + C x 2 arsinh ( a x ) d x = x 3 arsinh ( a x ) 3 ( a 2 x 2 2 ) a 2 x 2 + 1 9 a 3 + C x m arsinh ( a x ) d x = x m + 1 arsinh ( a x ) m + 1 a m + 1 x m + 1 a 2 x 2 + 1 d x ( m 1 ) arsinh ( a x ) 2 d x = 2 x + x arsinh ( a x ) 2 2 a 2 x 2 + 1 arsinh ( a x ) a + C arsinh ( a x ) n d x = x arsinh ( a x ) n n a 2 x 2 + 1 arsinh ( a x ) n 1 a + n ( n 1 ) arsinh ( a x ) n 2 d x arsinh ( a x ) n d x = x arsinh ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a 2 x 2 + 1 arsinh ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) arsinh ( a x ) n + 2 d x ( n 1 , 2 )

## Inverse hyperbolic cosine integration formulas

arcosh ( a x ) d x = x arcosh ( a x ) a x + 1 a x 1 a + C x arcosh ( a x ) d x = x 2 arcosh ( a x ) 2 arcosh ( a x ) 4 a 2 x a x + 1 a x 1 4 a + C x 2 arcosh ( a x ) d x = x 3 arcosh ( a x ) 3 ( a 2 x 2 + 2 ) a x + 1 a x 1 9 a 3 + C x m arcosh ( a x ) d x = x m + 1 arcosh ( a x ) m + 1 a m + 1 x m + 1 a x + 1 a x 1 d x ( m 1 ) arcosh ( a x ) 2 d x = 2 x + x arcosh ( a x ) 2 2 a x + 1 a x 1 arcosh ( a x ) a + C arcosh ( a x ) n d x = x arcosh ( a x ) n n a x + 1 a x 1 arcosh ( a x ) n 1 a + n ( n 1 ) arcosh ( a x ) n 2 d x arcosh ( a x ) n d x = x arcosh ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a x + 1 a x 1 arcosh ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) arcosh ( a x ) n + 2 d x ( n 1 , 2 )

## Inverse hyperbolic tangent integration formulas

artanh ( a x ) d x = x artanh ( a x ) + ln ( 1 a 2 x 2 ) 2 a + C x artanh ( a x ) d x = x 2 artanh ( a x ) 2 artanh ( a x ) 2 a 2 + x 2 a + C x 2 artanh ( a x ) d x = x 3 artanh ( a x ) 3 + ln ( 1 a 2 x 2 ) 6 a 3 + x 2 6 a + C x m artanh ( a x ) d x = x m + 1 artanh ( a x ) m + 1 a m + 1 x m + 1 1 a 2 x 2 d x ( m 1 )

## Inverse hyperbolic cotangent integration formulas

arcoth ( a x ) d x = x arcoth ( a x ) + ln ( a 2 x 2 1 ) 2 a + C x arcoth ( a x ) d x = x 2 arcoth ( a x ) 2 arcoth ( a x ) 2 a 2 + x 2 a + C x 2 arcoth ( a x ) d x = x 3 arcoth ( a x ) 3 + ln ( a 2 x 2 1 ) 6 a 3 + x 2 6 a + C x m arcoth ( a x ) d x = x m + 1 arcoth ( a x ) m + 1 + a m + 1 x m + 1 a 2 x 2 1 d x ( m 1 )

## Inverse hyperbolic secant integration formulas

arsech ( a x ) d x = x arsech ( a x ) 2 a arctan 1 a x 1 + a x + C x arsech ( a x ) d x = x 2 arsech ( a x ) 2 ( 1 + a x ) 2 a 2 1 a x 1 + a x + C x 2 arsech ( a x ) d x = x 3 arsech ( a x ) 3 1 3 a 3 arctan 1 a x 1 + a x x ( 1 + a x ) 6 a 2 1 a x 1 + a x + C x m arsech ( a x ) d x = x m + 1 arsech ( a x ) m + 1 + 1 m + 1 x m ( 1 + a x ) 1 a x 1 + a x d x ( m 1 )

## Inverse hyperbolic cosecant integration formulas

arcsch ( a x ) d x = x arcsch ( a x ) + 1 a arcoth 1 a 2 x 2 + 1 + C x arcsch ( a x ) d x = x 2 arcsch ( a x ) 2 + x 2 a 1 a 2 x 2 + 1 + C x 2 arcsch ( a x ) d x = x 3 arcsch ( a x ) 3 1 6 a 3 arcoth 1 a 2 x 2 + 1 + x 2 6 a 1 a 2 x 2 + 1 + C x m arcsch ( a x ) d x = x m + 1 arcsch ( a x ) m + 1 + 1 a ( m + 1 ) x m 1 1 a 2 x 2 + 1 d x ( m 1 )

## References

List of integrals of inverse hyperbolic functions Wikipedia

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