Supriya Ghosh (Editor)

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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