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SolveAPowerOverAPowerOfALogarithm​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable)

 Method (no overloads)

Summary

A power of x over a whole power of an affine function of a logarithm,
x^p (A + B L)^(-k) with L = ln(c x^r), onto the exponential integral. With
t = A + B L and L' = s/x, dx = x dt/(B s), and
x^(p + 1) e^(-(p + 1) t/(B s)) is a constant, so the integral is that constant over
B s times int e^((p + 1) t/(B s)) t^(-k) dt, which is Ei by parts. A
polynomial beside it is summed a power at a time, and the logarithm of a power of a linear
is the same question under u = d + e x. Rubi's 3.1.2 and 3.3,
(d x)^m/(a + b ln(c x^n))^k; 1/ln(x) is Ei(ln x), which is li(x).
https://github.com/asc-community/AngouriMath/issues/1501

Remarks

The constant is written as SolveAPowerTimesAHalfOddPowerOfTheLogarithm(AngouriMath.Entity,AngouriMath.Entity.Variable) writes it, and for ln(x) itself it is e^(-(p + 1) A/B), wherever ln(x) is real. Beside 1/x the rate is 0 and the integral is a power of t, which the
rules for logarithms answer.

























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