Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Cosine of a x/Logarithm Form

Theorem

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

\displaystyle \frac 1 {2 a p \sqrt{p^2 + q^2} } \ln \left\vert{ \frac {p \tanh a x + \sqrt{p^2 + q^2} } {p \tanh a x - \sqrt{p^2 + q^2} } }\right\vert + C & : \text{condition to be established} \\ \displaystyle \frac 1 {a p \sqrt{p^2 + q^2} } \arctan \frac {p \tanh a x} {\sqrt{p^2 + q^2} } + C & : \text{condition to be established} \\ \end{cases}$

Also see

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