AngouriMath
SolveASymbolicPowerOfAHyperbolicFunctionBesideAnOddPowerOfTheOther(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
A symbolic power of sinh(y) beside an odd whole power of cosh(y) , under
u = sinh(y) , where cosh(y)^(2j + 1) dy is (1 + u^2)^j du ; and a
symbolic power ofcosh(y) beside an odd power of sinh(y) , under
u = cosh(y) , with (u^2 - 1)^j . cosh(a + b x)^3 sinh(a + b x)^n is
(1 + u^2) u^n/b . A symbolic power of sech(y) or csch(y) the same way
underu = sech(y) or u = csch(y) , with tanh(y) and coth(y) beside it read as quotients: sech(a + b x)^n tanh(a + b x)^3 is
sinh^3 cosh^(-3) sech^n , which is -(1 - u^2) u^(n - 1)/b .
https://github.com/asc-community/AngouriMath/issues/233
symbolic power of
under
https://github.com/asc-community/AngouriMath/issues/233
Remarks
Rubi's four in 6.7.1 were past the budget:
SolveByBiochesOddHyperbolicSubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) reads whole and rational powers, and
a symbolic one is neither. Exact:d/dx F(sinh(y)) = y' cosh(y) F'(sinh(y)) wherever
F is differentiable, whatever the power, and (1 + sinh(y)^2)^j = cosh(y)^(2j) for a whole j . Odd powers from three: the first is the derivative of a power beside
it, which the rules for that answer. So were itssech(a + b x)^n tanh(a + b x)^3 and
coth(a + b x)^3 csch(a + b x)^n . The reciprocal is the variable, not its reciprocal
to the power negated:csch(y)^n is e^(2 pi i n) sinh(y)^(-n) where the sine is
negative, andu^n under u = csch(y) is csch(y)^n everywhere.
SolveByBiochesOddHyperbolicSubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) reads whole and rational powers, and
a symbolic one is neither. Exact:
it, which the rules for that answer. So were its
to the power negated:
negative, and
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