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

Theorem

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

Proof
We have:


 * $\dfrac \d {\d x} \paren {q \sinh a x + p \cosh a x} = a q \cosh a x + a p \sinh a x$

Thus:

Also see

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


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


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


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


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