Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x plus 1

Theorem

 * $\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x + 1} } = \frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} + \frac 1 {2 a \paren {\cosh a x + 1} } + C$

Proof
Let:

Then: