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

Theorem

 * $\ds \int \frac {\d x} {p^2 - q^2 \sinh^2 a x} = \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {p + \sqrt {p^2 + q^2} \tanh a x} {p - \sqrt {p^2 + q^2} \tanh a x} } + C$

Proof
Then:

Also see

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