Primitive of Reciprocal of Power of Hyperbolic Cosine of a x

Theorem

$\ds \int \frac {\d x} {\cosh^n a x} = \frac {\sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C$

Proof

\(\ds \int \frac {\d x} {\cosh^n a x}\)

\(=\)

\(\ds \int \sech^n a x \rd x\)

Definition 2 of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C\)

Primitive of $\sech^n a x$

\(\ds \)

\(=\)

\(\ds \frac {\tanh a x} {a \paren {n - 1} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C\)

Definition 2 of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \frac {\sinh a x} {a \cosh a x \paren {n - 1} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C\)

Definition 2 of Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \frac {\sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C\)

simplifying

$\blacksquare$

Also see

• Primitive of $\dfrac {\d x} {\sinh^n a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.588$