\[ \int \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \log \left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \polylog \left (2, 1+\frac {\left (-a d +b c \right ) \left (f x +e \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right )}{-a d +b c}-\frac {\polylog \left (3, 1+\frac {\left (-a d +b c \right ) \left (f x +e \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right )}{-a d +b c} \]
command
int(ln(e*(d*x+c)/(b*x+a))*ln((a*d-b*c)*(f*x+e)/(-c*f+d*e)/(b*x+a))/(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (-\frac {\frac {e \left (d x +c \right ) a f}{b x +a}-\frac {e^{2} \left (d x +c \right ) b}{b x +a}-c e f +d \,e^{2}}{e \left (c f -e d \right )}\right )}{2 a d -2 c b}-\frac {a f \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{2 \left (a d -c b \right ) \left (a f -b e \right )}-\frac {a f \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \polylog \left (2, \frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{\left (a d -c b \right ) \left (a f -b e \right )}+\frac {a f \polylog \left (3, \frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{\left (a d -c b \right ) \left (a f -b e \right )}+\frac {b e \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{2 \left (a d -c b \right ) \left (a f -b e \right )}+\frac {b e \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \polylog \left (2, \frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{\left (a d -c b \right ) \left (a f -b e \right )}-\frac {b e \polylog \left (3, \frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )}{\left (a d -c b \right ) \left (a f -b e \right )}\) | \(524\) |
Maple 2021.1 output
\[ \int \frac {\ln \left (\frac {\left (d x +c \right ) e}{b x +a}\right ) \ln \left (\frac {\left (a d -b c \right ) \left (f x +e \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]________________________________________________________________________________________