AngouriMath
SolveAPairOfHyperbolicPowersTwoApart(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
A pair of hyperbolic powers two apart whose coefficients kill the reduction's
residual:cosh(y)^p - (p - 1)/p cosh(y)^(p - 2) is
sinh(y) cosh(y)^(p - 1)/p , and sinh(y)^p + (p - 1)/p sinh(y)^(p - 2) is
cosh(y) sinh(y)^(p - 1)/p . Neither power has an elementary antiderivative on
its own for a fractionalp , and the combination does, which is why the terms
must not be split apart before this is asked.
residual:
its own for a fractional
must not be split apart before this is asked.
Remarks
From the reduction int cosh^p = sinh cosh^(p - 1)/p + (p - 1)/p int cosh^(p - 2) ,
with the coefficient of the lower power chosen so that what is left to integrate is
nothing; differentiating the answer throughsinh^2 = cosh^2 - 1 is the whole
proof. Rubi's 6.5.1 and 6.6.1 --x/sech(x)^(3/2) - x sqrt(sech(x))/3 is x times such a pair -- meet it beside a polynomial, which the common factor of a sum
gathers and parts then separates.
https://github.com/asc-community/AngouriMath/issues/718
with the coefficient of the lower power chosen so that what is left to integrate is
nothing; differentiating the answer through
proof. Rubi's 6.5.1 and 6.6.1 --
gathers and parts then separates.
https://github.com/asc-community/AngouriMath/issues/718
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