AngouriMath
SolveSecantPowersOfALogarithmTwoApartAsTheDerivativeOfAProduct(AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
A power of the variable times a secant or cosecant power of a logarithm, beside the
power two below it, where the pair is the derivative of one product. With
u = a + b ln(c x^n) , so that u' = B/x for B = b n , and k = m + 1 ,
d/dx [x^k sec(u)^(p - 2) (B (p - 2) tan(u) - k)] = x^m (D sec(u)^p - E sec(u)^(p - 2)) with D = B^2 (p - 1)(p - 2) and E = B^2 (p - 2)^2 + k^2 , so
x^m (alpha sec(u)^p + beta sec(u)^(p - 2)) integrates to alpha/D times the
product whereverbeta D + alpha E = 0 . The cosecant's product is
-x^k csc(u)^(p - 2) (B (p - 2) cot(u) + k) , with the same D and E .
power two below it, where the pair is the derivative of one product. With
product wherever
Remarks
Rubi's reduction for x^m sec(a + b ln(c x^n))^p lowers p by two and leaves
E/D times the lower power, which alone is a hypergeometric function; where
E is zero -- b^2 n^2 (p - 2)^2 + (m + 1)^2 = 0 -- or the lower power is the
integrand's other term, the product is the whole answer. Declined before: the
exponentials are written for a numerical imaginaryb only, and
sec(a + i ln(c x^n)/(n (p - 2)))^p , x^m sec(a + 2 ln(c x^(sqrt(-(1 + m)^2)/2)))^3 and -(1 + b^2 n^2) sec(u) + 2 b^2 n^2 sec(u)^3 were unanswered or ran out of time.
The derivative uses onlyz^(p - 2) z^2 = z^p , x^m x = x^k and
ln(c x^n)' = n/x , which hold for principal powers wherever the base is not zero,
so the answer holds wherever the integrand is defined. Not fork = 0 , where the
logarithm's own derivative is beside it and the substitution answers. Rubi's 4.7.5.
https://github.com/asc-community/AngouriMath/issues/233
integrand's other term, the product is the whole answer. Declined before: the
exponentials are written for a numerical imaginary
The derivative uses only
so the answer holds wherever the integrand is defined. Not for
logarithm's own derivative is beside it and the substitution answers. Rubi's 4.7.5.
https://github.com/asc-community/AngouriMath/issues/233
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