Primitive of Cube of Hyperbolic Cosecant of a x

Theorem

$\ds \int \csch^3 a x \rd x = \frac {-\csch a x \coth a x} {2 a} - \frac 1 {2 a} \ln \size {\tanh a x} + C$

Proof

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

\(=\)

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

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

\(\ds \)

\(=\)

\(\ds \frac {\csch a x \coth a x} {2 a} - \ln \size {\tanh a x} + C\)

Primitive of $\csch a x$

$\blacksquare$

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csch a x$: $14.638$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(33)$ Integrals Involving $\csch a x$: $17.33.3.$