Helpful Hyperbolic Identities, Basic Hyperbolic Function Definitions, Derivatives of Hyperbolic Functions, Integrals of Hyperbolic Functions, Derivatives of Inverse Hyperbolic Functions, ★ Integrals of Inverse Hyperbolic Functions ★

Odd identity for sinh

sinh(−x) = −sinh x

Even identity for cosh

cosh(−x) = cosh x

Fundamental hyperbolic identity

cosh²x − sinh²x = 1

Tanh-sech identity

1 − tanh²x = sech²x

Double angle identity (sinh)

sinh(2x) = 2 sinh x cosh x

Double angle identity (cosh)

cosh(2x) = cosh²x + sinh²x

sinh

sinh x = (eˣ − e⁻ˣ) / 2

cosh

cosh x = (eˣ + e⁻ˣ) / 2

tanh

tanh x = sinh x / cosh x

csch

csch x = 1 / sinh x

sech

sech x = 1 / cosh x

coth

coth x = cosh x / sinh x

cosh x

Derivative of sinh x

sinh x

Derivative of cosh x

sech²x

Derivative of tanh x

−csch x coth x

Derivative of csch x

−sech x tanh x

Derivative of sech x

−csch²x

Derivative of coth x

Integral of sinh x

cosh x + C

Integral of cosh x

sinh x + C

Integral of sech²x

tanh x + C

Integral of csch²x

−coth x + C

Integral of sech x tanh x

−sech x + C

Integral of csch x coth x

−csch x + C

Derivative of sinh⁻¹x

1 / √(1 + x²)

Derivative of cosh⁻¹x

1 / √(x² − 1)

Derivative of tanh⁻¹x

1 / (1 − x²)

Derivative of csch⁻¹x

−1 / (|x|√(x² + 1))

Derivative of sech⁻¹x

−1 / (x√(1 − x²))

Derivative of coth⁻¹x

1 / (1 − x²)

Integral of 1/√(a² + x²)

sinh⁻¹(x/a) + C

Integral of 1/√(x² − a²)

cosh⁻¹(x/a) + C

Integral of 1/(x√(a² − x²))

−(1/a) sech⁻¹(x/a) + C

Integral of 1/(x√(a² + x²))

−(1/a) csch⁻¹(|x|/a) + C

Integral of 1/(a² − x²), x² < a²

(1/a) tanh⁻¹(x/a) + C

Integral of 1/(a² − x²), x² > a²

(1/a) coth⁻¹(x/a) + C