Primitive of Product of Hyperbolic Secant and Tangent

Theorem

$\ds \int \sech x \tanh x \rd x = -\sech x + C$

where $C$ is an arbitrary constant.

Proof

From Derivative of Hyperbolic Secant:

$\dfrac \d {\d x} \sech x = -\sech x \tanh x$

The result follows from the definition of primitive.

$\blacksquare$

Sources

• 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxxi)}$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.37$
• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $23$