37.1 Problem number 216

$$\int \frac{{\cot }^5(e+fx)}{a+b{\tan }^2(e+fx)}dx$$

Optimal antiderivative $$\frac{(a+b)({\cot }^2(fx+e))}{2a^{2}f}-\frac{{\cot }^4(fx+e)}{4af}+\frac{\ln \left(\cos (fx+e)\right)}{(a-b)f}+\frac{(a^2+ab+b^2)\ln \left(\tan (fx+e)\right)}{a^{3}f}+\frac{b^3\ln (a+b({\tan }^2(fx+e)))}{2a^3(a-b)f}$$

command

integrate(cot(f*x+e)**5/(a+b*tan(f*x+e)**2),x)

Sympy 1.10.1 under Python 3.10.4 output

$$\text{Timed out}$$

Sympy 1.8 under Python 3.8.8 output

$$\text{output too large to display}$$