Primitive of Hyperbolic Secant Function

Theorem

Arcsine Form

$\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$

Arctangent of Hyperbolic Sine Form

$\ds \int \sech x \rd x = \map \arctan {\sinh x} + C$

Arctangent of Exponential Form

$\ds \int \sech x \rd x = 2 \map \arctan {e^x} + C$

Arctangent of Half Hyperbolic Tangent Form

$\ds \int \sech x \rd x = 2 \map \arctan {\tanh \dfrac x 2} + C$

Also see

• Primitive of Hyperbolic Sine Function
• Primitive of Hyperbolic Cosine Function

• Primitive of Hyperbolic Tangent Function
• Primitive of Hyperbolic Cotangent Function

• Primitive of Hyperbolic Cosecant Function

Sources

• 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxvii)}$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.29$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.29.$