Primitive of Inverse Hyperbolic Cotangent of x over a over x

Theorem

\(\ds \int \dfrac 1 x \arcoth \dfrac x a \rd x\)

\(=\)

\(\ds -\sum_{k \mathop \ge 0} \frac 1 {\paren {2 k + 1}^2} \paren {\frac a x}^{2 k + 1} + C\)

\(\ds \)

\(=\)

\(\ds -\paren {\frac a x + \frac {\paren {a / x}^3} {3^2} + \frac {\paren {a / x}^5} {5^2} + \frac {\paren {a / x}^7} {7^2} + \cdots} + C\)

Proof

\(\ds \arcoth \dfrac x a\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \paren {\frac a x}^{2 n + 1}\)

Power Series Expansion for Real Area Hyperbolic Cotangent

\(\ds \leadsto \ \ \)

\(\ds \frac 1 x \arcoth \dfrac x a\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} a^{2 n + 1} \frac 1 {x^{2 n + 2} }\)

\(\ds \leadsto \ \ \)

\(\ds \int \frac 1 x \arcoth \dfrac x a \rd x\)

\(=\)

\(\ds \int \paren {\sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} a^{2 n + 1} \frac 1 {x^{2 n + 2} } } \rd x\)

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \int \paren {\frac 1 {2 n + 1} a^{2 n + 1} \frac 1 {x^{2 n + 2} } \rd x}\)

Fubini's Theorem

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} a^{2 n + 1} \frac {-1} {2 n + 1}\frac 1 {x^{2 n + 1} } + C\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds -\sum_{k \mathop \ge 0} \frac 1 {\paren {2 k + 1}^2} \paren {\frac a x}^{2 k + 1} + C\)

rearranging

$\blacksquare$

Also see

• Primitive of $\dfrac 1 x \arsinh \dfrac x a$

• Primitive of $\dfrac 1 x \arcosh \dfrac x a$

• Primitive of $\dfrac 1 x \artanh \dfrac x a$

• Primitive of $\dfrac 1 x \arsech \dfrac x a$

• Primitive of $\dfrac 1 x \arcsch \dfrac x a$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.664$