sqrt(c + d tan(x))/(a + i a tan(x))^2 is answered in 250,000 characters that the default precision cannot evaluate

[Rafael-SOWNet]

sqrt(c + d*tan(x))/(a + i*a*tan(x))^2 integrates 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 = 107/100 and x at 57/100 and 129/100, 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 as 65536 d^572 256 256 16 16 ... and d^892, and terms that large cancel.

Under u = tan(x) it is sqrt(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 under t = sqrt(c + d u) the question is 2 d^3 t^2/((t^2 - c - i d)^3 (t^2 - c + i d)); the swell is in its answer. With a number for a the integral is declined, and with 1 + u under 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))^2 for k = 1, 3 and 5. Seen on 85d6601c and on 2b86349c.

Part of #233.

[Rafael-SOWNet]

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.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[Rafael-SOWNet]

The three defects behind these sizes are fixed, each in its own PR: #1795 puts constants holding i in 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.