Primitive of Cube of Hyperbolic Secant of a x

Theorem

$\ds \int \sech^3 a x \rd x = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C$

Proof

\(\ds \int \sech^3 x \rd x\)

\(=\)

\(\ds \frac {\sech a x \tanh a x} {2 a} + \frac 1 2 \int \sech a x \rd x\)

Primitive of $\sech^n a x$ where $n = 3$

\(\ds \)

\(=\)

\(\ds \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C\)

Primitive of $\sech a x$

$\blacksquare$

Also see

• Primitive of $\tanh^3 a x$
• Primitive of $\coth^3 a x$
• Primitive of $\csch^3 a x$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sech a x$: $14.628$