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

Theorem

 * $\ds \int \frac {\d x} {p + q \sinh a x} = \frac 1 {a \sqrt{p^2 + q^2} } \ln \size {\frac {q e^{a x} + p - \sqrt {p^2 + q^2} } {q e^{a x} + p + \sqrt {p^2 + q^2} } } + C$

Proof
Let:

Hence:

The discriminant of $q u^2 + 2 p u - q$ is given by:

Hence the sign of $\map {\operatorname {Disc} } {q u^2 + 2 p u - q}$ is always positive.

So:

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


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