Primitive of Inverse Hyperbolic Cosecant of x over a over x

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Theorem

\(\ds \int \dfrac 1 x \arcsch \dfrac x a \rd x\) \(=\) \(\ds \begin {cases} \ds \frac 1 2 \map \ln {\dfrac x a} \map \ln {\dfrac {4 a} x} - \sum_{n \mathop \ge 0} \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C & : 0 < x < a \\ \ds \frac 1 2 \map \ln {\dfrac {-x} a} \map \ln {\dfrac {-x} {4 a} } + \sum_{n \mathop \ge 0} \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C & : -a < x < 0 \\ \ds \sum_{n \mathop \ge 0} \frac {\paren {-1}^{n + 1} \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac a x}^{2 n + 1} + C & : \size x > a \\ \end {cases}\)
\(\ds \) \(\) \(\ds \)
\(\ds \) \(=\) \(\ds \begin {cases} \dfrac 1 2 \map \ln {\dfrac x a} \map \ln {\dfrac {4 a} x} + \dfrac 1 {2 \times 2^2} \paren {\dfrac x a}^2 - \dfrac {1 \times 3} {2 \times 4 \times 4^2} \paren {\dfrac x a}^4 + \dotsb + C & : 0 < x < a \\ \dfrac 1 2 \map \ln {\dfrac {-x} a} \map \ln {\dfrac {-x} {4 a} } - \dfrac 1 {2 \times 2^2} \paren {\dfrac x a}^2 + \dfrac {1 \times 3} {2 \times 4 \times 4^2} \paren {\dfrac x a}^4 - \dotsb + C & : -a < x < 0 \\ -\dfrac a x + \dfrac 1 {2 \times 3^2} \paren {\dfrac a x}^3 - \dfrac {1 \times 3} {2 \times 4 \times 5^2} \paren {\dfrac a x}^5 + \dotsb + C & : \size x > a \\ \end {cases}\)


Proof

For $\size x < a$:

\(\ds \arcsch \dfrac x a\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac a x}^{2 n + 1}\) Power Series Expansion for Real Area Hyperbolic Cosecant
\(\ds \leadsto \ \ \) \(\ds \frac 1 x \arcsch \dfrac x a\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } a^{2 n + 1} \frac 1 {x^{2 n + 2} }\)
\(\ds \leadsto \ \ \) \(\ds \int \frac 1 x \arcsch \dfrac x a \rd x\) \(=\) \(\ds \int \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } a^{2 n + 1} \frac 1 {x^{2 n + 2} } }\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {\int \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } a^{2 n + 1} \frac 1 {x^{2 n + 2} } \rd x}\) Fubini's Theorem
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \frac {a^{2 n + 1} } {\paren {-\paren {2 n + 1} } } \frac 1 {x^{2 n + 1} } + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n + 1} \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac a x}^{2 n + 1} + C\) rearranging

$\Box$


For $0 < x < a$:

\(\ds \arcsch \dfrac x a\) \(=\) \(\ds \ln \dfrac {2 a} x - \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \paren {\frac x a}^{2 n} }\) Power Series Expansion for Real Area Hyperbolic Cosecant
\(\ds \leadsto \ \ \) \(\ds \frac 1 x \arcsch \dfrac x a\) \(=\) \(\ds \dfrac 1 x \ln \frac {2 a} x - \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \frac 1 {a^{2 n} } x^{2 n - 1} }\)
\(\ds \leadsto \ \ \) \(\ds \int \frac 1 x \arcsch \dfrac x a \rd x\) \(=\) \(\ds \int \paren {\dfrac 1 x \ln \frac {2 a} x - \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \frac 1 {a^{2 n} } x^{2 n - 1} } } \rd x\)
\(\ds \) \(=\) \(\ds \int -\frac 1 x \ln \frac x {2 a} \rd x - \paren {\sum_{n \mathop = 0}^\infty \int \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \frac 1 {a^{2 n} } x^{2 n - 1} \rd x}\) Fubini's Theorem and Logarithm of Reciprocal
\(\ds \) \(=\) \(\ds \frac 1 2 \map {\ln^2} {\dfrac x {2 a} } - \paren {\sum_{n \mathop = 0}^\infty \int \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \frac 1 {a^{2 n} } x^{2 n - 1} \rd x} + C\) Primitive of $\dfrac {\ln x} x$: Corollary
\(\ds \) \(=\) \(\ds \frac 1 2 \map {\ln^2} {\dfrac x {2 a} } - \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 n \paren {2 n} } \frac 1 {a^{2 n} } \dfrac {x^{2 n} } {2 n} } + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac 1 2 \map {\ln^2} {\dfrac x {2 a} } - \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C\) rearranging
\(\ds \) \(=\) \(\ds \frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} + \map {\ln^2} 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C\) Lemma
\(\ds \) \(=\) \(\ds \frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} - \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C\) subsuming $\map {\ln^2} 2$ into the constant of integration

$\Box$


For $-a < x < 0$:

\(\ds \arcsch \dfrac x a\) \(=\) \(\ds -\ln \dfrac {-2 a} x + \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \paren {\frac x a}^{2 n} }\) Power Series Expansion for Real Area Hyperbolic Cosecant
\(\ds \leadsto \ \ \) \(\ds \frac 1 x \arcsch \dfrac x a\) \(=\) \(\ds -\frac 1 x \ln \frac {-2 x} a + \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \frac 1 {a^{2 n} } x^{2 n - 1} }\)
\(\ds \leadsto \ \ \) \(\ds \int \frac 1 x \arcsch \dfrac x a \rd x\) \(=\) \(\ds \int \frac 1 x \ln \frac {-x} {2 a} \rd x + \paren {\sum_{n \mathop = 0}^\infty \int \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \frac 1 {a^{2 n} } x^{2 n - 1} \rd x}\) Fubini's Theorem, as above
\(\ds \) \(=\) \(\ds -\frac 1 2 \map {\ln^2} {\dfrac {-x} {2 a} } + \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C\) from above
\(\ds \) \(=\) \(\ds -\frac 1 2 \map \ln {\dfrac {-a} x} \map \ln {\dfrac {-4 a} x} + \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C\) from above

$\blacksquare$


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