\(\int (-\frac {1}{(a+b x) (a-c+(b-d) x) \log (\frac {a+b x}{c+d x})}+\frac {\log (1-\frac {c+d x}{a+b x})}{(a+b x) (c+d x) \log ^2(\frac {a+b x}{c+d x})}) \, dx\) [75]

Optimal result

Integrand size = 88, antiderivative size = 45 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(b c-a d) \log \left (\frac {a+b x}{c+d x}\right )} \]

[Out]

-ln(1+(-d*x-c)/(b*x+a))/(-a*d+b*c)/ln((b*x+a)/(d*x+c))

Rubi [F]

\[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx \]

[In]

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]

[Out]

-Defer[Int][1/((a + b*x)*(a - c + (b - d)*x)*Log[(a + b*x)/(c + d*x)]), x] + (b*Defer[Int][Log[1 - (c + d*x)/(
a + b*x)]/((a + b*x)*Log[(a + b*x)/(c + d*x)]^2), x])/(b*c - a*d) - (d*Defer[Int][Log[1 - (c + d*x)/(a + b*x)]
/((c + d*x)*Log[(a + b*x)/(c + d*x)]^2), x])/(b*c - a*d)

Rubi steps \begin{align*} \text {integral}& = -\int \frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )} \, dx+\int \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 )} \, dx \\ & = -\int \frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )} \, dx+\int \left (\frac {b \log \left (1-\frac {c+d x}{a+b x}\right )}{(b c-a d) (a+b x) \log ^2\left (\frac {a+b x}{c+d x}\right )}-\frac {d \log \left (1-\frac {c+d x}{a+b x}\right )}{(b c-a d) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx \\ & = \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 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(b c-a d) \log \left (\frac {a+b x}{c+d x}\right )} \]

[In]

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]

[Out]

-(Log[1 - (c + d*x)/(a + b*x)]/((b*c - a*d)*Log[(a + b*x)/(c + d*x)]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(47)=94\).

Time = 220.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.89

method	result	size
parallelrisch	\(-\frac {2 \ln \left (\frac {b x -d x +a -c}{b x +a}\right ) b^{3} d^{3}-\ln \left (\frac {b x -d x +a -c}{b x +a}\right ) b^{4} d^{2}-\ln \left (\frac {b x -d x +a -c}{b x +a}\right ) b^{2} d^{4}}{\ln \left (\frac {b x +a}{d x +c}\right ) \left (b -d \right )^{2} \left (a d -c b \right ) b^{2} d^{2}}\)	\(130\)
risch	\(\frac {2 i \ln \left (b x -d x +a -c \right )}{\left (a d -c b \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 (a d -c b \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 )}\)	\(503\)

[In]

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,method=_RETURNVERBOSE)

[Out]

-(2*ln((b*x-d*x+a-c)/(b*x+a))*b^3*d^3-ln((b*x-d*x+a-c)/(b*x+a))*b^4*d^2-ln((b*x-d*x+a-c)/(b*x+a))*b^2*d^4)/ln(
(b*x+a)/(d*x+c))/(b-d)^2/(a*d-b*c)/b^2/d^2

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (\frac {{\left (b - d\right )} x + a - c}{b x + a}\right )}{{\left (b c - a d\right )} \log \left (\frac {b x + a}{d x + c}\right )} \]

[In]

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")

[Out]

-log(((b - d)*x + a - c)/(b*x + a))/((b*c - a*d)*log((b*x + a)/(d*x + c)))

Sympy [F(-2)]

Exception generated. \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left ({\left (b - d\right )} x + a - c\right ) - \log \left (b x + a\right )}{{\left (b c - a d\right )} \log \left (b x + a\right ) - {\left (b c - a d\right )} \log \left (d x + c\right )} \]

[In]

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")

[Out]

-(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))

Giac [F]

\[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\int { -\frac {1}{{\left ({\left (b - d\right )} x + a - c\right )} {\left (b x + a\right )} \log \left (\frac {b x + a}{d x + c}\right )} + \frac {\log \left (-\frac {d x + c}{b x + a} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}} \,d x } \]

[In]

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")

[Out]

integrate(-1/(((b - d)*x + a - c)*(b*x + a)*log((b*x + a)/(d*x + c))) + log(-(d*x + c)/(b*x + a) + 1)/((b*x +
a)*(d*x + c)*log((b*x + a)/(d*x + c))^2), x)

Mupad [B] (verification not implemented)

Time = 1.57 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \left (-\frac {1}{(a+b x) (a-c+(b-d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\frac {\ln \left (1-\frac {c+d\,x}{a+b\,x}\right )}{\ln \left (\frac {a+b\,x}{c+d\,x}\right )\,\left (a\,d-b\,c\right )} \]

[In]

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)

[Out]

log(1 - (c + d*x)/(a + b*x))/(log((a + b*x)/(c + d*x))*(a*d - b*c))