AngouriMath
SolveAPowerTimesATrigonometricOfALogarithm(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
A power of the variable times a sine or cosine of a logarithm, in closed form:
int x^m sin(L) dx is x^(m + 1)((m + 1) sin(L) - B cos(L))/((m + 1)^2 + B^2) whenever L' = B/x for a constant B , which a + b ln(c x^n) is with
B = b n ; the cosine's is the same with (m + 1) cos(L) + B sin(L) .
Remarks
Two rounds of parts close on the integrand -- the sine's remainder is the cosine's
integral and the cosine's is the sine's -- and the pair is solved rather than
iterated. Differentiating the answer is the whole proof: the cross terms cancel and
what is left isx^m sin(L)((m + 1)^2 + B^2) . The hyperbolic twin needs no rule,
the library writingsinh as exponentials that fold against the logarithm;
the sine and cosine are nodes and fold against nothing. Rubi's 4.7.5.
https://github.com/asc-community/AngouriMath/issues/718
integral and the cosine's is the sine's -- and the pair is solved rather than
iterated. Differentiating the answer is the whole proof: the cross terms cancel and
what is left is
the library writing
the sine and cosine are nodes and fold against nothing. Rubi's 4.7.5.
https://github.com/asc-community/AngouriMath/issues/718
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