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

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

[Out]

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

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Rubi [F]  time = 0.515643, 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 = {} \[ \int \left (\frac{1}{(c+d x) (-a+c+(-b+d) x) \log \left (\frac{a+b x}{c+d x}\right )}+\frac{\log \left (1-\frac{a+b x}{c+d x}\right )}{(a+b x) (c+d x) \log ^2\left (\frac{a+b x}{c+d x}\right )}\right ) \, dx \]

Verification is Not applicable to the result.

[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*} \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}{(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]  time = 0.311511, size = 44, normalized size = 0.98 \[ \frac{\log \left (1-\frac{a+b x}{c+d x}\right )}{(a d-b c) \log \left (\frac{a+b x}{c+d x}\right )} \]

Antiderivative was successfully verified.

[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)])

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Maple [C]  time = 1.322, size = 662, normalized size = 14.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[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*csgn(I*(b*x+a)/(d*x+c))*csgn(I*(b*x+a))*csgn(I/(d*x+c))*Pi+I*csgn(I*(b*x+a)/(d*x+c))^2*c
sgn(I*(b*x+a))*Pi+I*csgn(I*(b*x+a)/(d*x+c))^2*csgn(I/(d*x+c))*Pi+I*Pi*csgn(I/(d*x+c))*csgn(I*(b*x-d*x+a-c))*cs
gn(I/(d*x+c)*(b*x-d*x+a-c))-I*Pi*csgn(I/(d*x+c))*csgn(I/(d*x+c)*(b*x-d*x+a-c))^2-I*csgn(I*(b*x+a)/(d*x+c))^3*P
i+2*I*Pi*csgn(I/(d*x+c)*(b*x-d*x+a-c))^2-I*Pi*csgn(I*(b*x-d*x+a-c))*csgn(I/(d*x+c)*(b*x-d*x+a-c))^2-I*Pi*csgn(
I/(d*x+c)*(b*x-d*x+a-c))^3-2*I*Pi+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 = 1.49692, size = 80, normalized size = 1.78 \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/(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))

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Fricas [A]  time = 2.06832, size = 104, normalized size = 2.31 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Sympy [A]  time = 3.05976, size = 44, normalized size = 0.98 \begin{align*} \frac{\log{\left (\frac{- a - b x}{c + d x} + 1 \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/(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]

log((-a - b*x)/(c + d*x) + 1)/(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 (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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)