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

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Theorem

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


Proof

\(\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x}\) \(=\) \(\ds \int \frac {\csch^2 a x \rd x} {p^2 \csch^2 a x + q^2 \coth^2 a x}\) multiplying numerator and denominator by $\csch^2 a x$
\(\ds \) \(=\) \(\ds \int \frac {\csch^2 a x \rd x} {p^2 \paren {\coth^2 a x - 1} + q^2 \coth^2 a x}\) Difference of Squares of Hyperbolic Cotangent and Cosecant
\(\ds \) \(=\) \(\ds \int \frac {\csch^2 a x \rd x} {\paren {p^2 + q^2} \coth^2 a x - p^2}\) simplifying

Let:

\(\ds u\) \(=\) \(\ds \coth a x\)
\(\ds \leadsto \ \ \) \(\ds \d u\) \(=\) \(\ds -a \csch^2 a x \rd x\) Derivative of Hyperbolic Cotangent Function
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x}\) \(=\) \(\ds \frac 1 a \int \frac {-\d u} {\paren {p^2 + q^2} u^2 - p^2}\) Integration by Substitution: substituting $u = \tanh a x$
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \frac 1 {a \paren {p^2 + q^2} } \int \frac {\rd u} {\frac {p^2} {p^2 + q^2} - u^2}\) rearranging into a standard form


We have:

\(\ds \int \frac {\d x} {p^2 + q^2 \cosh^2 a x}\) \(=\) \(\ds \frac 1 {a \paren {p^2 + q^2} } \int \frac {\rd u} {\paren {\frac p {\sqrt {p^2 + q^2} } }^2 - u^2}\) from $(1)$
\(\ds \) \(=\) \(\ds \frac 1 {a \paren {p^2 + q^2} } \frac {\sqrt {p^2 + q^2} } {2 p} \ln \size {\frac {\frac p {\sqrt {p^2 + q^2} } + u} {\frac p {\sqrt {p^2 + q^2} } - u} } + C\) Primitive of $\dfrac 1 {a^2 - x^2}$
\(\ds \) \(=\) \(\ds \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {\frac p {\sqrt {p^2 + q^2} } + \coth a x} {\frac p {\sqrt {p^2 + q^2} } - \coth a x} } + C\) substituting $u = \coth a x$
\(\ds \) \(=\) \(\ds \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {\frac p {\sqrt {p^2 + q^2} } + \frac 1 {\tanh a x} } {\frac p {\sqrt {p^2 + q^2} } - \frac 1 {\tanh a x} } } + C\) Definition of Real Hyperbolic Cotangent
\(\ds \) \(=\) \(\ds \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {p \tanh a x + \sqrt {p^2 + q^2} } {p \tanh a x - \sqrt {p^2 + q^2} } } + C\) multiplying numerator and denominator of argument of $\ln$ by $\sqrt {p^2 + q^2} \tanh a x$

$\blacksquare$


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