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