Primitive of Inverse Hyperbolic Sine of x over a over x

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Theorem

\(\ds \int \dfrac 1 x \arsinh \dfrac x a \rd x\) \(=\) \(\ds \begin {cases} \ds \sum_{n \mathop \ge 0} \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac x a}^{2 n + 1} + C & : \size x < a \\ \ds \frac 1 2 \map {\ln^2} {\dfrac {2 x} a} + \sum_{n \mathop \ge 0} \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C & : x > a \\ \ds -\frac 1 2 \map {\ln^2} {\dfrac {-2 x} a} + \sum_{n \mathop \ge 0} \frac {\paren {-1}^{n + 1} \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C & : x < -a \end {cases}\)
\(\ds \) \(\) \(\ds \)
\(\ds \) \(=\) \(\ds \begin {cases} \dfrac x a - \dfrac 1 {2 \times 3^2} \paren {\dfrac x a}^3 + \dfrac {1 \times 3} {2 \times 4 \times 5^2} \paren {\dfrac x a}^5 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 7^2} \paren {\dfrac x a}^7 \dotsb + C & : \size x < a \\ \ds \frac 1 2 \map {\ln^2} {\dfrac {2 x} a} - \dfrac 1 {2 \times 2^2} \paren {\dfrac a x}^2 + \dfrac {1 \times 3} {2 \times 4 \times 4^2} \paren {\dfrac a x}^4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 5^2} \paren {\dfrac a x}^6 + \dotsb + C & : x > a \\ \ds -\frac 1 2 \map {\ln^2} {\dfrac {-2 x} a} + \dfrac 1 {2 \times 2^2} \paren {\dfrac a x}^2 - \dfrac {1 \times 3} {2 \times 4 \times 4^2} \paren {\dfrac a x}^4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 5^2} \paren {\dfrac a x}^6 - \dotsb + C & : x < -a \end {cases}\)


Proof

For $\size x < a$:

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

$\Box$


For $x > a$:

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

$\Box$


For $x < a$:

\(\ds \arsinh \dfrac x a\) \(=\) \(\ds -\ln \paren {-2 x} + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n + 1} \paren {n!}^2 n x^{2 n} } }\) Power Series Expansion for Real Area Hyperbolic Sine
\(\ds \leadsto \ \ \) \(\ds \frac 1 x \arsinh \dfrac x a\) \(=\) \(\ds -\frac 1 x \ln \frac {-2 x} a + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n + 1} \paren {n!}^2 n} a^{2 n} \paren {\dfrac 1 x}^{2 n + 1} }\)
\(\ds \leadsto \ \ \) \(\ds \int \frac 1 x \arsinh \dfrac x a \rd x\) \(=\) \(\ds -\int \frac 1 x \ln \frac {2 x} a \rd x + \paren {\sum_{n \mathop = 1}^\infty \int \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n + 1} \paren {n!}^2 n} a^{2 n} \paren {\dfrac 1 x}^{2 n + 1} \rd x}\) from above
\(\ds \) \(=\) \(\ds -\frac 1 2 \map {\ln^2} {\dfrac {-2 x} a} - \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C\) from above
\(\ds \) \(=\) \(\ds -\frac 1 2 \map {\ln^2} {\dfrac {-2 x} a} + \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n + 1} \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C\) from above

$\blacksquare$


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