\(\int (\frac {1}{(c+d x) (-a+c+(-b+d) x) \log (\frac {a+b x}{c+d x})}+\frac {\log (1-\frac {a+b x}{c+d x})}{(a+b x) (c+d x) \log ^2(\frac {a+b x}{c+d x})}) \, dx\) [74]

Optimal result

Integrand size = 87, antiderivative size = 45 \[ \int \left (\frac {1}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(b c-a d) \log \left (\frac {a+b x}{c+d x}\right )} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

Defer[Int][1/((c + d*x)*(-a + c + (-b + d)*x)*Log[(a + b*x)/(c + d*x)]), x] + (b*Defer[Int][Log[1 - (a + b*x)/
(c + d*x)]/((a + b*x)*Log[(a + b*x)/(c + d*x)]^2), x])/(b*c - a*d) - (d*Defer[Int][Log[1 - (a + b*x)/(c + d*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}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac {a+b x}{c+d x}\right )} \, dx+\int \frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )} \, dx \\ & = \int \frac {1}{(c+d 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 {a+b x}{c+d 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 {a+b x}{c+d 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 {a+b x}{c+d 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 {a+b x}{c+d x}\right )}{(c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )} \, dx}{b c-a d}+\int \frac {1}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac {a+b x}{c+d x}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \left (\frac {1}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(-b c+a d) \log \left (\frac {a+b x}{c+d x}\right )} \]

[In]

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

[Out]

Log[1 - (a + b*x)/(c + d*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. \(132\) vs. \(2(47)=94\).

Time = 218.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.96

method	result	size
parallelrisch	\(-\frac {-\ln \left (-\frac {b x -d x +a -c}{d x +c}\right ) b^{2} d^{4}+2 \ln \left (-\frac {b x -d x +a -c}{d x +c}\right ) b^{3} d^{3}-\ln \left (-\frac {b x -d x +a -c}{d x +c}\right ) b^{4} d^{2}}{\ln \left (\frac {b x +a}{d x +c}\right ) \left (b -d \right )^{2} d^{2} \left (a d -c b \right ) b^{2}}\)	\(133\)

[In]

int(1/(d*x+c)/(-a+c+(-b+d)*x)/ln((b*x+a)/(d*x+c))+ln(1+(-b*x-a)/(d*x+c))/(b*x+a)/(d*x+c)/ln((b*x+a)/(d*x+c))^2
,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.11 \[ \int \left (\frac {1}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (-\frac {{\left (b - d\right )} x + a - c}{d x + c}\right )}{{\left (b c - a d\right )} \log \left (\frac {b x + a}{d x + c}\right )} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

integrate(1/(d*x+c)/(-a+c+(-b+d)*x)/ln((b*x+a)/(d*x+c))+ln(1+(-b*x-a)/(d*x+c))/(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 = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.31 \[ \int \left (\frac {1}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=-\frac {\log \left (-{\left (b - d\right )} x - a + c\right ) - \log \left (b x + a\right )}{{\left (b c - a d\right )} \log \left (b x + a\right ) - {\left (b c - a d\right )} \log \left (d x + c\right )} \]

[In]

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

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

[In]

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

Mupad [B] (verification not implemented)

Time = 1.76 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \left (\frac {1}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac {a+b x}{c+d x}\right )}+\frac {\log \left (1-\frac {a+b x}{c+d x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac {a+b x}{c+d x}\right )}\right ) \, dx=\frac {\ln \left (1-\frac {a+b\,x}{c+d\,x}\right )}{\ln \left (\frac {a+b\,x}{c+d\,x}\right )\,\left (a\,d-b\,c\right )} \]

[In]

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

[Out]

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