\[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (\left (-d x +c \right ) \left (d x +c \right )^{2}\right ) 2^{\frac {2}{3}}}{8 c d}-\frac {3 \ln \left (d \left (-d x +c \right )+2^{\frac {2}{3}} d \left (d^{3} x^{3}-c^{3}\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{8 c d}+\frac {\arctan \left (\frac {\left (1-\frac {2^{\frac {1}{3}} \left (-d x +c \right )}{\left (d^{3} x^{3}-c^{3}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, 2^{\frac {2}{3}}}{4 c d} \]
command
Integrate[1/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{-\frac {1}{2}} \left (2 i \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt [3]{2} \left (3+i \sqrt {3}\right ) c+\sqrt [3]{2} \left (-3-i \sqrt {3}\right ) d x+2 i \sqrt {3} \sqrt [3]{-c^3+d^3 x^3}}{6 \sqrt [3]{-c^3+d^3 x^3}}\right )+2 \log \left (\sqrt {c} \sqrt {d} \left (-c+i \sqrt {3} c+d x-i \sqrt {3} d x+2\ 2^{2/3} \sqrt [3]{-c^3+d^3 x^3}\right )\right )-\log \left (-c d \left (\left (1+i \sqrt {3}\right ) c^2+\left (1+i \sqrt {3}\right ) d^2 x^2-2 (-2)^{2/3} d x \sqrt [3]{-c^3+d^3 x^3}-4 \sqrt [3]{2} \left (-c^3+d^3 x^3\right )^{2/3}+2 c \left (\left (-1-i \sqrt {3}\right ) d x+(-2)^{2/3} \sqrt [3]{-c^3+d^3 x^3}\right )\right )\right )\right )}{4 c d} \]
Mathematica 12.3 output
\[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx \]________________________________________________________________________________________