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SolveAsTheDerivativeOfAQuotientOverASumOfExponentials(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)

Method (no overloads)

Summary

The derivative of N/D, for D a sum of exponentials of x and constants and
N a constant plus a constant times each exponential in D, read off the
integrand over D^2: alone, or beside x by parts. Rubi's
x (b - a sinh(x))/(a + b sinh(x))^2 is x G' for G = -cosh(x)/(a + b sinh(x)),
so x G - integral G, and the integral of G is -ln(a + b sinh(x))/b.

Remarks

The exponentials' span is closed under the derivative, so (N/D)' = (N' D - N D')/D^2 with N' D - N D' linear in N's constants, and matching it to the integrand's
numerator is a linear system: each exponential a whole power of the first one, w,
and the identity read at six values of w. The answer is differentiated back at sampled
points.
The split took the numerator's terms apart, and the exponential substitution answered each
over the square of a quadratic in e^x with a piecewise on its discriminant; beside x,
parts integrated that piecewise again and ran past the budget, where the trigonometric
twin is SolveAsTheDerivativeOfASineRaisedFromNothingBesidePowersOfTheCosine(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)'s.
https://github.com/asc-community/AngouriMath/issues/233

AngouriMath

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AngouriMath

SolveAsTheDerivativeOfAQuotientOverASumOfExponentials​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

SolveAsTheDerivativeOfAQuotientOverASumOfExponentials(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)