37.2 Problem number 106

\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx \]

Optimal antiderivative \[ -\frac {\ln \left (\frac {\left (-a f +b e \right ) \left (d x +c \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right )^{2} \ln \left (1-\frac {\left (-a f +b e \right ) \left (d x +c \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right )}{-a f +b e}-\frac {2 \ln \left (\frac {\left (-a f +b e \right ) \left (d x +c \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right ) \polylog \left (2, \frac {\left (-a f +b e \right ) \left (d x +c \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right )}{-a f +b e}+\frac {2 \polylog \left (3, \frac {\left (-a f +b e \right ) \left (d x +c \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right )}{-a f +b e} \]

command

integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a)/(f*x+e),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ \frac {\log \left (\frac {a d f x + a c f - {\left (b d x + b c\right )} e}{b c f x + a c f - {\left (b d x + a d\right )} e}\right )^{2} \log \left (\frac {{\left (b c - a d\right )} f x + {\left (b c - a d\right )} e}{b c f x + a c f - {\left (b d x + a d\right )} e}\right ) + 2 \, {\rm Li}_2\left (-\frac {{\left (b c - a d\right )} f x + {\left (b c - a d\right )} e}{b c f x + a c f - {\left (b d x + a d\right )} e} + 1\right ) \log \left (\frac {a d f x + a c f - {\left (b d x + b c\right )} e}{b c f x + a c f - {\left (b d x + a d\right )} e}\right ) - 2 \, {\rm polylog}\left (3, \frac {a d f x + a c f - {\left (b d x + b c\right )} e}{b c f x + a c f - {\left (b d x + a d\right )} e}\right )}{a f - b e} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {\log \left (\frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right )^{2}}{b f x^{2} + a e + {\left (b e + a f\right )} x}, x\right ) \]