Primitive of Inverse Hyperbolic Secant of x over a over x/Corollary

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Theorem

\(\ds \int \dfrac 1 x \paren {-\sech^{-1} \dfrac x a} \rd x\) \(=\) \(\ds \frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} + \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} + \dfrac 1 {2 \times 2^2} \paren {\dfrac x a}^2 - \dfrac {1 \times 3} {2 \times 4 \times 4^2} \paren {\dfrac x a}^4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6^2} \paren {\dfrac x a}^6 + C\)

where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction.


Proof

\(\ds -\sech^{-1} \frac x a\) \(=\) \(\ds -\arsech \frac x a\) Definition of Real Inverse Hyperbolic Secant
\(\ds \leadsto \ \ \) \(\ds \int \dfrac 1 x \paren {-\sech^{-1} \dfrac x a} \rd x\) \(=\) \(\ds -\int \dfrac 1 x \arsech \dfrac x a \rd x\)
\(\ds \) \(=\) \(\ds -\paren {-\frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} - \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C}\) Primitive of $\dfrac 1 x \arsech \dfrac x a$
\(\ds \) \(=\) \(\ds \frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} + \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C\) Definition of Real Inverse Hyperbolic Secant

$\blacksquare$


Sources

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