\[ \int \frac {c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \]
Optimal antiderivative \[ -\frac {\ln \left (d x +c \right )}{d}+\frac {3 \ln \left (d \left (d x +2 c \right )-d \left (d^{3} x^{3}+2 c^{3}\right )^{\frac {1}{3}}\right )}{2 d}-\frac {\arctan \left (\frac {\left (1+\frac {2 d x +4 c}{\left (d^{3} x^{3}+2 c^{3}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{d} \]
command
Integrate[(c - d*x)/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{2 c^3+d^3 x^3}}{4 c+2 d x+\sqrt [3]{2 c^3+d^3 x^3}}\right )}{d}+\frac {\log \left (-2 c-d x+\sqrt [3]{2 c^3+d^3 x^3}\right )}{d}-\frac {\log \left (4 c^2+4 c d x+d^2 x^2+(2 c+d x) \sqrt [3]{2 c^3+d^3 x^3}+\left (2 c^3+d^3 x^3\right )^{2/3}\right )}{2 d} \]
Mathematica 12.3 output
\[ \int \frac {c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \]________________________________________________________________________________________