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# Combinations of logarithms, rational and trigonometric functions

DOI link for Combinations of logarithms, rational and trigonometric functions

Combinations of logarithms, rational and trigonometric functions book

# Combinations of logarithms, rational and trigonometric functions

DOI link for Combinations of logarithms, rational and trigonometric functions

Combinations of logarithms, rational and trigonometric functions book

## ABSTRACT

G :=

(−1)k (2k + 1)2

. (9.1.7)

In this paper we concentrate on integrals of the type (9.1.1) where the logarithm appears to the first power and the poles of the rational function are either real or purely imaginary. The method of partial fractions and scaling of the independent variable show that such integrals are linear combinations of

hn,1(b) :=

ln t dt

(1 + t)n , (9.1.8)

and

hn,2(b) :=

ln t dt

(1 + t2)n . (9.1.9)

The function hn,1 was evaluated in [64], where it was denoted simply by h. We complete this evaluation in Section 9.4, by identifying a polynomial defined in [64]. The closed-form of hn,1 involves the Stirling numbers of the first kind. The evaluation of hn,2 is discussed in Section 9.6. The value of hn,2 involves the tangent integral

Ti2(x) :=

tan−1 t t

. (9.1.10)

The case of integrals with more complicated pole structure will be described in a future publication.