Primitive of Reciprocal of Hyperbolic Tangent of a x

Theorem

$\ds \int \frac {\d x} {\tanh a x} = \frac {\ln \size {\sinh a x} } a + C$

Proof

\(\ds \int \frac {\d x} {\tanh a x}\)

\(=\)

\(\ds \int \coth a x \rd x\)

Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \frac {\ln \size {\sinh a x} } a + C\)

Primitive of $\coth a x$

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\sinh a x}$

• Primitive of $\dfrac 1 {\cosh a x}$

• Primitive of $\dfrac 1 {\coth a x}$

• Primitive of $\dfrac 1 {\sech a x}$

• Primitive of $\dfrac 1 {\csch a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tanh a x$: $14.609$