The numerical computation of two transcendental functions related to the exponential integral

Algorithms for the computation of numerical values of the two transcendental functions \[ ∫ 0 x 1 t [ Ei ⁡ ( t ) − γ − ln ⁡ | t | ] d t and ∫ 0 x 1 t [ Ei ⁡ ( t ) − γ − ln ⁡ | t | ] exp ⁡ ( − t ) d t , ∫ _0^x {1/t} [Ei (t) - γ - ln | t |]dt {and} ∫ _0^x {1/t} [Ei (t) - γ - ln | t |]exp ( - t)dt, \] where γ γ is Euler’s constant and Ei ⁡ ( t ) Ei (t) is the exponential integral, are presented for all ranges of the real variable x x . A table of values of these functions is also given.