\[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {d \ln \! \left (\tan \! \left (f x +e \right )\right )}{6 c^{\frac {2}{3}} f}+\frac {d \ln \! \left (c^{\frac {1}{3}}-\left (c +d \tan \! \left (f x +e \right )\right )^{\frac {1}{3}}\right )}{2 c^{\frac {2}{3}} f}-\frac {d \arctan \! \left (\frac {\left (c^{\frac {1}{3}}+2 \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 c^{\frac {1}{3}}}\right ) \sqrt {3}}{3 c^{\frac {2}{3}} f}+\frac {x \left (c -\sqrt {-d^{2}}\right )^{\frac {1}{3}}}{4}+\frac {d \ln \! \left (\cos \! \left (f x +e \right )\right ) \left (c -\sqrt {-d^{2}}\right )^{\frac {1}{3}}}{4 f \sqrt {-d^{2}}}+\frac {3 d \ln \! \left (\left (c -\sqrt {-d^{2}}\right )^{\frac {1}{3}}-\left (c +d \tan \! \left (f x +e \right )\right )^{\frac {1}{3}}\right ) \left (c -\sqrt {-d^{2}}\right )^{\frac {1}{3}}}{4 f \sqrt {-d^{2}}}-\frac {d \arctan \! \left (\frac {\left (1+\frac {2 \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}}{\left (c -\sqrt {-d^{2}}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, \left (c -\sqrt {-d^{2}}\right )^{\frac {1}{3}}}{2 f \sqrt {-d^{2}}}+\frac {x \left (c +\sqrt {-d^{2}}\right )^{\frac {1}{3}}}{4}-\frac {d \ln \! \left (\cos \! \left (f x +e \right )\right ) \left (c +\sqrt {-d^{2}}\right )^{\frac {1}{3}}}{4 f \sqrt {-d^{2}}}-\frac {3 d \ln \! \left (\left (c +\sqrt {-d^{2}}\right )^{\frac {1}{3}}-\left (c +d \tan \! \left (f x +e \right )\right )^{\frac {1}{3}}\right ) \left (c +\sqrt {-d^{2}}\right )^{\frac {1}{3}}}{4 f \sqrt {-d^{2}}}+\frac {d \arctan \! \left (\frac {\left (1+\frac {2 \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}}{\left (c +\sqrt {-d^{2}}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, \left (c +\sqrt {-d^{2}}\right )^{\frac {1}{3}}}{2 f \sqrt {-d^{2}}}-\frac {\cot \! \left (f x +e \right ) \left (c +d \tan \! \left (f x +e \right )\right )^{\frac {1}{3}}}{f} \]
command
integrate(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {Timed out} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ -\frac {{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}}{f \tan \left (f x + e\right )} \]