\[ \int \left (\frac {1}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx \]
Optimal antiderivative \[ -\frac {\ln \! \left (1+\frac {-b x -a}{d x +c}\right )}{\left (-a d +b c \right ) \ln \! \left (\frac {b x +a}{d x +c}\right )} \]
command
int(1/(d*x+c)/(-a+c+(-b+d)*x)/ln((b*x+a)/(d*x+c))+ln(1+(-b*x-a)/(d*x+c))/(b*x+a)/(d*x+c)/ln((b*x+a)/(d*x+c))^2,x)
Maple 2022.1 output
\[\int \frac {1}{\left (d x +c \right ) \left (-a +c +\left (-b +d \right ) x \right ) \ln \left (\frac {b x +a}{d x +c}\right )}+\frac {\ln \left (1+\frac {-b x -a}{d x +c}\right )}{\left (b x +a \right ) \left (d x +c \right ) \ln \left (\frac {b x +a}{d x +c}\right )^{2}}\, dx\]
Maple 2021.1 output
\[ \frac {2 i \ln \left (b x -d x +a -c \right )}{\left (a d -b c \right ) \left (\pi \,\mathrm {csgn}\left (i \left (b x +a \right )\right ) \mathrm {csgn}\left (\frac {i}{d x +c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )-\pi \,\mathrm {csgn}\left (i \left (b x +a \right )\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{d x +c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{3}+2 i \ln \left (b x +a \right )-2 i \ln \left (d x +c \right )\right )}-\frac {-i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )\right ) \mathrm {csgn}\left (\frac {i}{d x +c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )+i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{d x +c}\right ) \mathrm {csgn}\left (i \left (b x -d x +a -c \right )\right ) \mathrm {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{d x +c}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{d x +c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{d x +c}\right ) \mathrm {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{d x +c}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (b x -d x +a -c \right )\right ) \mathrm {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{d x +c}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{d x +c}\right )^{3}+2 i \pi \mathrm {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{d x +c}\right )^{2}+2 \ln \left (b x +a \right )-2 i \pi }{\left (a d -b c \right ) \left (-i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )\right ) \mathrm {csgn}\left (\frac {i}{d x +c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )+i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{d x +c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{3}+2 \ln \left (b x +a \right )-2 \ln \left (d x +c \right )\right )} \]