I am trying to understand how WolframAlpha reduces the following trig expression $$ \frac{\ln \left(\sin \left(\frac{\alpha x}{2}\right)+\cos \left(\frac{\alpha x}{2}\right)\right)}{\alpha }-\frac{\ln \left(\cos \left(\frac{\alpha x}{2}\right)-\sin \left(\frac{\alpha x}{2}\right)\right)}{\alpha }$$ into $$\frac{\ln (\tan (\alpha x)+\sec (\alpha x))}{\alpha }$$
I tried using TrigExpand, TrigReduce and FullSimplify with no use.
CODE:
FullSimplify[Integrate[Sec[\[Alpha]*x], x], Assumptions -> {\[Alpha] > 0 && Element[x, Reals]}]
Integrate[Sec[α x], x] /. {x -> 1, α -> 3.}
is different fromLog[Tan[α x] + Sec[α x]]/α /. {x -> 1, α -> 3.}
, so the functions are not the same. If that's not important, maybeFullSimplify[Integrate[Sec[α x], x], TransformationFunctions -> {Automatic, Log@*Exp}]
suffice? $\endgroup$