Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Sine of a x

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {p^2 + q^2 \sinh^2 a x} = \begin{cases}

\displaystyle \frac 1 {a p \sqrt{q^2 - p^2} } \arctan \frac {\sqrt{q^2 - p^2} \tanh a x} p + C & : p^2 < q^2 \\ \displaystyle \frac 1 {2 a p \sqrt{p^2 - q^2} } \ln \left\vert{ \frac {p + \sqrt{p^2 - q^2} \tanh a x} {p - \sqrt{p^2 - q^2} \tanh a x} }\right\vert + C & : p^2 > q^2 \\ \end{cases}$

Also see

 * Primitive of $\dfrac 1 {p^2 + q^2 \cosh^2 a x}$