Integrand size = 88, antiderivative size = 45 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(b c-a d) \log \left (\frac {a+b x}{c+d x}\right )} \] Output:
-ln(1-(d*x+c)/(b*x+a))/(-a*d+b*c)/ln((b*x+a)/(d*x+c))
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(b c-a d) \log \left (\frac {a+b x}{c+d x}\right )} \] Input:
Integrate[-(1/((a + b*x)*(a - c + (b - d)*x)*Log[(a + b*x)/(c + d*x)])) + Log[1 - (c + d*x)/(a + b*x)]/((a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)] ^2),x]
Output:
-(Log[1 - (c + d*x)/(a + b*x)]/((b*c - a*d)*Log[(a + b*x)/(c + d*x)]))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}-\frac {1}{(a+b x) (a+x (b-d)-c) \log \left (\frac {a+b x}{c+d x}\right )}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \int \frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) \log ^2\left (\frac {a+b x}{c+d x}\right )}dx}{b c-a d}-\frac {d \int \frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}dx}{b c-a d}-\int \frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}dx\) |
Input:
Int[-(1/((a + b*x)*(a - c + (b - d)*x)*Log[(a + b*x)/(c + d*x)])) + Log[1 - (c + d*x)/(a + b*x)]/((a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]^2),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.23 (sec) , antiderivative size = 503, normalized size of antiderivative = 11.18
\[\frac {2 i \ln \left (b x -d x +a -c \right )}{\left (d a -b c \right ) \left (\pi \,\operatorname {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right ) \operatorname {csgn}\left (i \left (b x +a \right )\right ) \operatorname {csgn}\left (\frac {i}{d x +c}\right )-\pi \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2} \operatorname {csgn}\left (i \left (b x +a \right )\right )-\pi \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2} \operatorname {csgn}\left (\frac {i}{d x +c}\right )+\pi \operatorname {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 \,\operatorname {csgn}\left (i \left (b x -d x +a -c \right )\right ) \operatorname {csgn}\left (\frac {i}{b x +a}\right ) \operatorname {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{b x +a}\right )-i \pi \,\operatorname {csgn}\left (i \left (b x -d x +a -c \right )\right ) \operatorname {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{b x +a}\right )^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{b x +a}\right ) \operatorname {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{b x +a}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (b x -d x +a -c \right )}{b x +a}\right )^{3}+2 \ln \left (b x +a \right )}{\left (d a -b c \right ) \left (-i \pi \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2} \operatorname {csgn}\left (i \left (b x +a \right )\right )+i \pi \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right )^{2} \operatorname {csgn}\left (\frac {i}{d x +c}\right )-i \pi \,\operatorname {csgn}\left (\frac {i \left (b x +a \right )}{d x +c}\right ) \operatorname {csgn}\left (i \left (b x +a \right )\right ) \operatorname {csgn}\left (\frac {i}{d x +c}\right )+2 \ln \left (b x +a \right )-2 \ln \left (d x +c \right )\right )}\]
Input:
int(-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)
Output:
2*I/(a*d-b*c)/(Pi*csgn(I*(b*x+a)/(d*x+c))*csgn(I*(b*x+a))*csgn(I/(d*x+c))- Pi*csgn(I*(b*x+a)/(d*x+c))^2*csgn(I*(b*x+a))-Pi*csgn(I*(b*x+a)/(d*x+c))^2* csgn(I/(d*x+c))+Pi*csgn(I*(b*x+a)/(d*x+c))^3+2*I*ln(b*x+a)-2*I*ln(d*x+c))* ln(b*x-d*x+a-c)-(I*Pi*csgn(I*(b*x-d*x+a-c))*csgn(I/(b*x+a))*csgn(I/(b*x+a) *(b*x-d*x+a-c))-I*Pi*csgn(I*(b*x-d*x+a-c))*csgn(I/(b*x+a)*(b*x-d*x+a-c))^2 -I*Pi*csgn(I/(b*x+a))*csgn(I/(b*x+a)*(b*x-d*x+a-c))^2+I*Pi*csgn(I/(b*x+a)* (b*x-d*x+a-c))^3+2*ln(b*x+a))/(a*d-b*c)/(-I*Pi*csgn(I*(b*x+a)/(d*x+c))^3+I *Pi*csgn(I*(b*x+a)/(d*x+c))^2*csgn(I*(b*x+a))+I*Pi*csgn(I*(b*x+a)/(d*x+c)) ^2*csgn(I/(d*x+c))-I*Pi*csgn(I*(b*x+a)/(d*x+c))*csgn(I*(b*x+a))*csgn(I/(d* x+c))+2*ln(b*x+a)-2*ln(d*x+c))
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (\frac {{\left (b - d\right )} x + a - c}{b x + a}\right )}{{\left (b c - a d\right )} \log \left (\frac {b x + a}{d x + c}\right )} \] Input:
integrate(-1/(b*x+a)/(a-c+(b-d)*x)/log((b*x+a)/(d*x+c))+log(1-(d*x+c)/(b*x +a))/(b*x+a)/(d*x+c)/log((b*x+a)/(d*x+c))^2,x, algorithm="fricas")
Output:
-log(((b - d)*x + a - c)/(b*x + a))/((b*c - a*d)*log((b*x + a)/(d*x + c)))
Exception generated. \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\text {Exception raised: TypeError} \] Input:
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)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left ({\left (b - d\right )} x + a - c\right ) - \log \left (b x + a\right )}{{\left (b c - a d\right )} \log \left (b x + a\right ) - {\left (b c - a d\right )} \log \left (d x + c\right )} \] Input:
integrate(-1/(b*x+a)/(a-c+(b-d)*x)/log((b*x+a)/(d*x+c))+log(1-(d*x+c)/(b*x +a))/(b*x+a)/(d*x+c)/log((b*x+a)/(d*x+c))^2,x, algorithm="maxima")
Output:
-(log((b - d)*x + a - c) - log(b*x + a))/((b*c - a*d)*log(b*x + a) - (b*c - a*d)*log(d*x + c))
\[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\int { -\frac {1}{{\left ({\left (b - d\right )} x + a - c\right )} {\left (b x + a\right )} \log \left (\frac {b x + a}{d x + c}\right )} + \frac {\log \left (-\frac {d x + c}{b x + a} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}} \,d x } \] Input:
integrate(-1/(b*x+a)/(a-c+(b-d)*x)/log((b*x+a)/(d*x+c))+log(1-(d*x+c)/(b*x +a))/(b*x+a)/(d*x+c)/log((b*x+a)/(d*x+c))^2,x, algorithm="giac")
Output:
integrate(-1/(((b - d)*x + a - c)*(b*x + a)*log((b*x + a)/(d*x + c))) + lo g(-(d*x + c)/(b*x + a) + 1)/((b*x + a)*(d*x + c)*log((b*x + a)/(d*x + c))^ 2), x)
Time = 26.57 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\frac {\ln \left (1-\frac {c+d\,x}{a+b\,x}\right )}{\ln \left (\frac {a+b\,x}{c+d\,x}\right )\,\left (a\,d-b\,c\right )} \] Input:
int(log(1 - (c + d*x)/(a + b*x))/(log((a + b*x)/(c + d*x))^2*(a + b*x)*(c + d*x)) - 1/(log((a + b*x)/(c + d*x))*(a + b*x)*(a - c + x*(b - d))),x)
Output:
log(1 - (c + d*x)/(a + b*x))/(log((a + b*x)/(c + d*x))*(a*d - b*c))
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\frac {\mathrm {log}\left (\frac {b x -d x +a -c}{b x +a}\right )}{\mathrm {log}\left (\frac {b x +a}{d x +c}\right ) \left (a d -b c \right )} \] Input:
int(-1/(b*x+a)/(a-c+(b-d)*x)/log((b*x+a)/(d*x+c))+log(1-(d*x+c)/(b*x+a))/( b*x+a)/(d*x+c)/log((b*x+a)/(d*x+c))^2,x)
Output:
log((a + b*x - c - d*x)/(a + b*x))/(log((a + b*x)/(c + d*x))*(a*d - b*c))