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

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {p + q \sinh a x} = \frac 1 {a \sqrt{p^2 + q^2} } \ln \left\vert{\frac {q e^{a x} + p - \sqrt {p^2 + q^2} } {q e^{a x} + p + \sqrt {p^2 + q^2} } }\right\vert + C$

Proof
The discriminant of $p u^2 + 2 q u - p$ is $4 \left({p^2 + q^2}\right)$.

Thus:

Also see

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


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