Primitive of Reciprocal of p plus q by Hyperbolic Cosine of a x

Theorem

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

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

Also see

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


 * Primitive of $\dfrac 1 {p + q \tanh a x}$


 * Primitive of $\dfrac 1 {p + q \coth a x}$


 * Primitive of $\dfrac 1 {q + p \operatorname{sech} a x}$


 * Primitive of $\dfrac 1 {q + p \operatorname{csch} a x}$