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

Optimal. Leaf size=112 \[ \frac{b \text{CannotIntegrate}\left (\frac{\log \left (1-\frac{c+d x}{a+b x}\right )}{(a+b x) \log ^2\left (\frac{a+b x}{c+d x}\right )},x\right )}{b c-a d}-\frac{d \text{CannotIntegrate}\left (\frac{\log \left (1-\frac{c+d x}{a+b x}\right )}{(c+d x) \log ^2\left (\frac{a+b x}{c+d x}\right )},x\right )}{b c-a d} \]

[Out]

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

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Rubi [A]  time = 0.483652, 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 = {} \[ \int \frac{\log \left (1-\frac{c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac{a+b x}{c+d x}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(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 \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 \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}\\ \end{align*}

Mathematica [A]  time = 0.574993, size = 0, normalized size = 0. \[ \int \frac{\log \left (1-\frac{c+d x}{a+b x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac{a+b x}{c+d x}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.008, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ( 1+{\frac{-dx-c}{bx+a}} \right ) \left ( \ln \left ({\frac{bx+a}{dx+c}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \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 )} - \int -\frac{1}{{\left ({\left (b^{2} - b d\right )} x^{2} + a^{2} - a c +{\left (a{\left (2 \, b - d\right )} - b c\right )} x\right )} \log \left (b x + a\right ) -{\left ({\left (b^{2} - b d\right )} x^{2} + a^{2} - a c +{\left (a{\left (2 \, b - d\right )} - b c\right )} x\right )} \log \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\frac{{\left (b - d\right )} x + a - c}{b x + a}\right )}{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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(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(log(-(d*x + c)/(b*x + a) + 1)/((b*x + a)*(d*x + c)*log((b*x + a)/(d*x + c))^2), x)