Integral of the secant function

The integral of the secant function of trigonometry was the subject of one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory. In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums. He wanted the solution for the purposes of cartography – specifically for constructing an accurate Mercator projection. In the 1640s, Henry Bond, a teacher of navigation, surveying, and other mathematical topics, compared Wright's numerically computed table of values of the integral of the secant with a table of logarithms of the tangent function, and consequently conjectured that

That conjecture became widely known, and in 1665, Isaac Newton was aware of it.

The problem was solved by Isaac Barrow. His proof of the result was the earliest use of partial fractions in integration. Adapted to modern notation, Barrow's proof began as follows:

Substituting ${\displaystyle u}$ for ${\displaystyle \sin \theta }$ reduces the integral to

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