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

Optimal. Leaf size=45 \[ -\frac{\log \left (1-\frac{c+d x}{a+b x}\right )}{(b c-a d) \log \left (\frac{a+b x}{c+d x}\right )} \]

[Out]

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

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Rubi [F]  time = 0.512104, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (-\frac{1}{(a+b x) (a-c+(b-d) x) \log \left (\frac{a+b x}{c+d x}\right )}+\frac{\log \left (1-\frac{c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac{a+b x}{c+d x}\right )}\right ) \, dx \]

Verification is Not applicable to the result.

[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*} \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}{(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]  time = 0.0851547, size = 45, normalized size = 1. \[ -\frac{\log \left (1-\frac{c+d x}{a+b x}\right )}{(b c-a d) \log \left (\frac{a+b x}{c+d x}\right )} \]

Antiderivative was successfully verified.

[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)]))

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Maple [C]  time = 1.214, size = 503, normalized size = 11.2 \begin{align*}{\frac{2\,i\ln \left ( bx-dx+a-c \right ) }{ad-bc} \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) }{dx+c}} \right ){\it csgn} \left ( i \left ( bx+a \right ) \right ){\it csgn} \left ({\frac{i}{dx+c}} \right ) \pi - \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ( bx+a \right ) \right ) \pi - \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{dx+c}} \right ) \pi + \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{3}\pi +2\,i\ln \left ( bx+a \right ) -2\,i\ln \left ( dx+c \right ) \right ) ^{-1}}-{\frac{1}{ad-bc} \left ( i\pi \,{\it csgn} \left ( i \left ( bx-dx+a-c \right ) \right ){\it csgn} \left ({\frac{i}{bx+a}} \right ){\it csgn} \left ({\frac{i \left ( bx-dx+a-c \right ) }{bx+a}} \right ) -i\pi \,{\it csgn} \left ( i \left ( bx-dx+a-c \right ) \right ) \left ({\it csgn} \left ({\frac{i \left ( bx-dx+a-c \right ) }{bx+a}} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ({\frac{i}{bx+a}} \right ) \left ({\it csgn} \left ({\frac{i \left ( bx-dx+a-c \right ) }{bx+a}} \right ) \right ) ^{2}+i\pi \, \left ({\it csgn} \left ({\frac{i \left ( bx-dx+a-c \right ) }{bx+a}} \right ) \right ) ^{3}+2\,\ln \left ( bx+a \right ) \right ) \left ( -i \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ( bx+a \right ) \right ) \pi +i \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{dx+c}} \right ) \pi -i{\it csgn} \left ({\frac{i \left ( bx+a \right ) }{dx+c}} \right ){\it csgn} \left ( i \left ( bx+a \right ) \right ){\it csgn} \left ({\frac{i}{dx+c}} \right ) \pi +2\,\ln \left ( bx+a \right ) -2\,\ln \left ( dx+c \right ) \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

2*I/(a*d-b*c)/(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*Pi+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*csgn(I*(b*x+a)/(d*x+c))^3*Pi+I*csgn(I*(b*x+a)/(d*x+c))^2*csgn(
I*(b*x+a))*Pi+I*csgn(I*(b*x+a)/(d*x+c))^2*csgn(I/(d*x+c))*Pi-I*csgn(I*(b*x+a)/(d*x+c))*csgn(I*(b*x+a))*csgn(I/
(d*x+c))*Pi+2*ln(b*x+a)-2*ln(d*x+c))

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Maxima [A]  time = 1.67998, size = 78, normalized size = 1.73 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]  time = 1.99166, size = 103, normalized size = 2.29 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Sympy [A]  time = 3.32851, size = 44, normalized size = 0.98 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)