\[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx \]
Optimal antiderivative \[ -\frac {\ln \! \left (1+\frac {-d x -c}{b x +a}\right )}{\left (-a d +b c \right ) \ln \! \left (\frac {b x +a}{d x +c}\right )} \]
command
integrate(-1/(b*x+a)/(a-c+(b-d)*x)/ln((b*x+a)/(d*x+c))+ln(1+(-d*x-c)/(b*x+a))/(b*x+a)/(d*x+c)/ln((b*x+a)/(d*x+c))**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {\log {\left (1 + \frac {- c - d x}{a + b x} \right )}}{a d \log {\left (\frac {a + b x}{c + d x} \right )} - b c \log {\left (\frac {a + b x}{c + d x} \right )}} \]