Primitive of Inverse Hyperbolic Secant of x over a over x

Theorem

 * $\displaystyle \int \frac {\sech^{-1} \dfrac x a \rd x} x = \begin {cases}

\displaystyle -\frac {\map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} } 2 - \sum_{k \mathop \ge 0} \frac {\paren {2 k + 1}!} {2^{2 k} \paren {k!}^2 \paren {2 k + 1}^3 \paren {2 k}^2} \paren {\frac x a}^{2 k} + C & : \sech^{-1} \dfrac x a > 0 \\ \displaystyle \frac {\map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} } 2 + \sum_{k \mathop \ge 0} \frac {\paren {2 k + 1}!} {2^{2 k} \paren {k!}^2 \paren {2 k + 1}^3 \paren {2 k}^2} \paren {\frac x a}^{2 k} + C & : \sech^{-1} \dfrac x a < 0 \\ \end {cases}$

Also see

 * Primitive of $\dfrac {\sinh^{-1} \frac x a} x$


 * Primitive of $\dfrac {\cosh^{-1} \frac x a} x$


 * Primitive of $\dfrac {\tanh^{-1} \frac x a} x$


 * Primitive of $\dfrac {\coth^{-1} \frac x a} x$


 * Primitive of $\dfrac {\csch^{-1} \frac x a} x$