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sqrt(c + d tan(x))/(a + i a tan(x))^2 is answered in 250,000 characters that the default precision cannot evaluate #1788
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- added a parent issue
on Oct 5, 2026 - changed the title
[-]sqrt(c + d tan(x))/(a + i a tan(x))^2 is integrated wrongly[/-][+]sqrt(c + d tan(x))/(a + i a tan(x))^2 is answered in 250,000 characters that the default precision cannot evaluate[/+]on Oct 5, 2026 Corrected the title and description: the answer is right -- it agrees with the integrand to 49 digits once evaluated with exact parameters and enough precision -- and what the first version reported as a wrong answer was the default-precision evaluation of its swollen coefficients. The defect is the answer's size.
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The three defects behind these sizes are fixed, each in its own PR: #1795 puts constants holding
iin lowest terms, #1797 writes a quadratic's discriminant one way so the powers of a quadratic share their arms, and #1799 lets the arm off the real line exclude the arms on the sign of the discriminant. Each PR's description has its rows measured before and after.
sqrt(c + d*tan(x))/(a + i*a*tan(x))^2integrates on master to an answer of about 250,000 characters, and evaluated at the default precision its derivative is not the integrand: it differs at every sample point. 2.5.0 declines it.The answer is right. With
a = 13/10,c = 7/10,d = 107/100andxat57/100and129/100, its derivative agrees with the integrand to 49 significant digits at 50, 300 and 2000 digits of precision. What fails at the default precision is the evaluation: the coefficients grow to products such as65536 d^572 256 256 16 16 ...andd^892, and terms that large cancel.Under
u = tan(x)it issqrt(c + d u)/((a + i a u)^2 (1 + u^2)), answered the same way. The partial fractions at the top find the root the two factors below the bar share,(i a)^2 (u - i)^3 (u + i), and undert = sqrt(c + d u)the question is2 d^3 t^2/((t^2 - c - i d)^3 (t^2 - c + i d)); the swell is in its answer. With a number forathe integral is declined, and with1 + uunder the root it is answered shortly.Rubi's 4.3.2.1 has three rows of this shape,
(c + d tan(x))^(k/2)/(a + i a tan(x))^2fork= 1, 3 and 5. Seen on85d6601cand on2b86349c.Part of #233.