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

Theorem

$\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x} = \dfrac 1 {a p \sqrt {p^2 + q^2} } \tanh^{-1} \dfrac {p \tanh a x} {\sqrt {p^2 + q^2} } + C$

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

$=$

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

$\ds $

$=$

$\ds \frac 1 {a p \sqrt {p^2 + q^2} } \tanh^{-1} \dfrac {p \tanh a x} {\sqrt {p^2 + q^2} } + C$

$\blacksquare$