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Answer an exponential of an arctangent where it is an exact derivative - #1938
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(e^(N arctan(a x)) (alpha + beta x) Q^s)' is c e^(N arctan(a x)) Q^(s - 1) times a quadratic in x, for Q = c + a^2 c x^2. The constant and squared coefficients of the integrand's quadratic give alpha and beta, and the linear one is the condition. e^(i n arctan(a x)) x^2 (c + a^2 c x^2)^(-1 - n^2/2) meets it and was left unevaluated (Rubi's 5.3.6:504). Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…AKING-CHANGES Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…-of-an-exponential-of-an-arctangent
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Part of #718.
e^(i n arctan(a x)) x^2 (c + a^2 c x^2)^(-1 - n^2/2)(Rubi's 5.3.6) was left unevaluated. No rule reads a polynomial beside the exponential and a symbolic power ofc + a^2 c x^2.The integrand is the derivative of
e^(N arctan(a x)) (α + β x) (c + a^2 c x^2)^(p + 1). Here N is the exponential's multiple,c + a^2 c x^2is the quadratic, and P is the quadratic polynomial beside it. Withs = p + 1, matching the coefficients of P fixes the two unknowns:It is an exact derivative where the middle coefficient also matches:
c (a N β + 2 s a^2 α) = p1.SolveAnExactDerivativeOfAnExponentialOfAnArctangentanswers that case. The candidate is checked by differentiating it back at sampled points.0f840669e^(i n atan(a x)) x^2 (c + a^2 c x^2)^(-1 - n^2/2)The answer is exact to 300 digits at four sets of constants, negative ones among them, with
npinned past the branch wrap.Measured:
53a0fa42, with 504 re-measured on0f840669. 504 is the one move.97ef01eb): 4,821 passed.AnExactDerivativeOfAnExponentialOfAnArctangentIntegralTest, two rows.BREAKING entry: An exponential of an arctangent beside a power of its quadratic is answered where it is an exact derivative. The 2.5.0 column was measured on the
v2.5.0package.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura