\[ \int \frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \polylog \left (2, 1+\frac {-a d +b c}{d \left (b x +a \right )}\right )}{-a d +b c}-\frac {\polylog \left (3, 1+\frac {-a d +b c}{d \left (b x +a \right )}\right )}{-a d +b c} \]
command
int(ln((a*d-b*c)/d/(b*x+a))*ln(e*(d*x+c)/(b*x+a))/(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (-\frac {\frac {e \left (d x +c \right ) b}{b x +a}-e d}{e d}\right )}{2 a d -2 c b}-\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )}{2 \left (a d -c b \right )}-\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \polylog \left (2, \frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )}{a d -c b}+\frac {\polylog \left (3, \frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )}{a d -c b}\) | \(186\) |
Maple 2021.1 output
\[ \int \frac {\ln \left (\frac {\left (d x +c \right ) e}{b x +a}\right ) \ln \left (\frac {a d -b c}{\left (b x +a \right ) d}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]________________________________________________________________________________________