Primitive of Reciprocal of Power of Hyperbolic Sine of a x

Theorem

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

for $n \ne 1$.

Proof

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

\(=\)

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

Definition 2 of Hyperbolic Cosecant

\(\ds \)

\(=\)

\(\ds \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x\)

Primitive of $\csch^n a x$

\(\ds \)

\(=\)

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

Definition 2 of Hyperbolic Cosecant

\(\ds \)

\(=\)

\(\ds \frac {-\cosh a x} {a \sinh a x \left({n - 1}\right) \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}\)

Definition 2 of Hyperbolic Cotangent

\(\ds \)

\(=\)

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

simplifying

We note that when $n = 1$, $\dfrac {n - 2} {n - 1}$ is undefined, rendering this derivation invalid.

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\sinh a x}$ for the case where $n = -1$

• Primitive of $\dfrac 1 {\cosh^n a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.560$