AngouriMath
SolveAPowerTimesAPowerOfTheLogarithm(AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
A power of x times a whole power of its logarithm, x^p ln(x)^n , by the
closed reduction
x^(p + 1) sum_(k = 0..n) (-1)^k n!/(n - k)! ln(x)^(n - k)/(p + 1)^(k + 1) , for
anyp but -1 , where it is ln(x)^(n + 1)/(n + 1) . And the same
forF = A + B ln(c x^r) in the logarithm's place, whose derivative is
s/x for s = B r : each step of parts brings a factor s , so the
sum is over(-s)^k n!/(n - k)! F^(n - k)/(p + 1)^(k + 1) , and at p = -1 it is F^(n + 1)/((n + 1) s) . Rubi's (e x)^q (a + b ln(c x^n))^3 , and
t^(-2 - m) (A + B ln(e t^n))^2 in the variable of the quotient substitution.
closed reduction
any
for
sum is over
Remarks
By parts n times, written out: each step takes one from the power of the
logarithm and divides byp + 1 . Timofeev's x^m ln(x)^2 and
ln(x)^2/x^(5/2) had no antiderivative -- the first step of by parts was
taken and the second, onx^m ln(x) with the symbol still in the exponent,
was not -- wherex^m ln(x) and x^2 ln(x)^2 were answered. Closed, exact,
and volunteered at any depth.
https://github.com/asc-community/AngouriMath/issues/718
logarithm and divides by
taken and the second, on
was not -- where
and volunteered at any depth.
https://github.com/asc-community/AngouriMath/issues/718
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