\[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx \]
Optimal antiderivative \[ \frac {\cot ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}-\frac {\cot ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )}{4 b n}+\frac {\ln \left (\sin \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b n} \]
command
integrate(cot(a+b*ln(c*x**n))**5/x,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: \left (a = 0 \vee a = - b \log {\left (c x^{n} \right )}\right ) \wedge \left (a = - b \log {\left (c x^{n} \right )} \vee b = 0\right ) \wedge \left (a = - b \log {\left (c x^{n} \right )} \vee n = 0\right ) \\\log {\left (x \right )} \cot ^{5}{\left (a \right )} & \text {for}\: b = 0 \\\log {\left (x \right )} \cot ^{5}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{2 b n} + \frac {\log {\left (\tan {\left (a + b \log {\left (c x^{n} \right )} \right )} \right )}}{b n} + \frac {1}{2 b n \tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}} - \frac {1}{4 b n \tan ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________