Newsgroups: sci.math.symbolic,sci.math
From: jgk@netcom.com (Joe Keane)
Subject: Re: Integrate[1 / Cos[t], {t, 0, ArcSin[s]}]
Message-ID: <jgkCxv7yM.Fwr@netcom.com>
Date: Tue, 18 Oct 1994 11:01:34 GMT

In article <37mtlp$7b1@news.ycc.yale.edu>
Richard Q. Chen  writes:
>I would like to simplify the output from Mma of the following integral
>       Integrate[1/Cos[t],{t,0,ArcSin[s]}]
>where I am interested only in real s with -1 < s < 1.
>By hand it only took me a few minutes to get the compact result
>       Log[(1+s)/(1-s)]/2

This is just arctanh s.  The integral of sec t is gd^-1 t, the inverse
Gudermannian function.  This function gives a neat mapping between circular
and hyperbolic functions, besides the usual one using imaginary numbers.

--
Joe Keane, amateur mathematician