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

Theorem

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

\dfrac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\dfrac {p \tanh a x + \sqrt {p^2 + q^2} } {p \tanh a x - \sqrt {p^2 + q^2} } } + C & : \text{condition to be established} \\ \dfrac 1 {a p \sqrt {p^2 + q^2} } \arctan \dfrac {p \tanh a x} {\sqrt {p^2 + q^2} } + C & : \text{condition to be established} \\ \end {cases}$

Also see

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