The following is a list of

indefinite integral In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...

s (

antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...

s) of expressions involving the

inverse hyperbolic function In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The ...

s. For a complete list of integral formulas, see lists of integrals. * In all formulas the constant is assumed to be nonzero, and denotes the

constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connect ...

. * 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

\int\operatorname(ax)\,dx=
  x\operatorname(ax)-\frac+C

\int x\operatorname(ax)\,dx=
  \frac+
  \frac-
  \frac+C

\int x^2\operatorname(ax)\,dx=
  \frac-
  \frac+C

\int x^m\operatorname(ax)\,dx=
  \frac-
  \frac\int\frac\,dx\quad(m\ne-1)

\int\operatorname(ax)^2\,dx=
  2x+x\operatorname(ax)^2-
  \frac+C

\int\operatorname(ax)^n\,dx=
  x\operatorname(ax)^n-
  \frac+
  n(n-1)\int\operatorname(ax)^\,dx

\int\operatorname(ax)^n\,dx=
  -\frac+
  \frac+
  \frac\int\operatorname(ax)^\,dx\quad(n\ne-1,-2)

Inverse hyperbolic cosine integration formulas

\int\operatorname(ax)\,dx=
  x\operatorname(ax)-
  \frac+C

\int x\operatorname(ax)\,dx=
  \frac-
  \frac-
  \frac+C

\int x^2\operatorname(ax)\,dx=
  \frac-\frac+C

\int x^m\operatorname(ax)\,dx=
  \frac-
  \frac\int\frac\,dx\quad(m\ne-1)

\int\operatorname(ax)^2\,dx=
  2x+x\operatorname(ax)^2-
  \frac+C

\int\operatorname(ax)^n\,dx=
  x\operatorname(ax)^n-
  \frac+
  n(n-1)\int\operatorname(ax)^\,dx

\int\operatorname(ax)^n\,dx=
  -\frac+
  \frac+
  \frac\int\operatorname(ax)^\,dx\quad(n\ne-1,-2)

Inverse hyperbolic tangent integration formulas

\int\operatorname(ax)\,dx=
  x\operatorname(ax)+
  \frac+C

\int x\operatorname(ax)\,dx=
  \frac-
  \frac+\frac+C

\int x^2\operatorname(ax)\,dx=
  \frac+
  \frac+\frac+C

\int x^m\operatorname(ax)\,dx=
  \frac-
  \frac\int\frac\,dx\quad(m\ne-1)

Inverse hyperbolic cotangent integration formulas

\int\operatorname(ax)\,dx=
  x\operatorname(ax)+
  \frac+C

\int x\operatorname(ax)\,dx=
  \frac-
  \frac+\frac+C

\int x^2\operatorname(ax)\,dx=
  \frac+
  \frac+\frac+C

\int x^m\operatorname(ax)\,dx=
  \frac+
  \frac\int\frac\,dx\quad(m\ne-1)

Inverse hyperbolic secant integration formulas

\int\operatorname(ax)\,dx=
  x\operatorname(ax)-
  \frac\operatorname\sqrt+C

\int x\operatorname(ax)\,dx=
  \frac-
  \frac\sqrt+C

\int x^2\operatorname(ax)\,dx=
  \frac-
  \frac\operatorname\sqrt-
  \frac\sqrt+C

\int x^m\operatorname(ax)\,dx=
  \frac+
  \frac\int\frac\,dx\quad(m\ne-1)

Inverse hyperbolic cosecant integration formulas

\int\operatorname(ax)\,dx=
  x\operatorname(ax)+
  \frac\operatorname\sqrt+C

\int x\operatorname(ax)\,dx=
  \frac+
  \frac\sqrt+C

\int x^2\operatorname(ax)\,dx=
  \frac-
  \frac\operatorname\sqrt+
  \frac\sqrt+C

\int x^m\operatorname(ax)\,dx=
  \frac+
  \frac\int\frac\,dx\quad(m\ne-1)

{{Lists of integrals Area functions Integrals of inverse hyperbolic functions