Optimal. Leaf size=46 \[ -\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}-\frac {\text {Li}_2\left (1-\frac {a}{a+b x}\right )}{b} \]
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Rubi [A] time = 0.16, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2488, 2411, 2343, 2333, 2315} \[ -\frac {\text {PolyLog}\left (2,1-\frac {a}{a+b x}\right )}{b}-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2333
Rule 2343
Rule 2411
Rule 2488
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx &=-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}+\frac {a \int \frac {\log \left (\frac {a}{a+b x}\right )}{x (a+b x)} \, dx}{b}\\ &=-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}+\frac {a \operatorname {Subst}\left (\int \frac {\log \left (\frac {a}{x}\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x\right )}{b^2}\\ &=-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}-\frac {a \operatorname {Subst}\left (\int \frac {\log (a x)}{\left (-\frac {a}{b}+\frac {1}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{b^2}\\ &=-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}-\frac {a \operatorname {Subst}\left (\int \frac {\log (a x)}{\frac {1}{b}-\frac {a x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{b^2}\\ &=-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}-\frac {\text {Li}_2\left (\frac {b x}{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 84, normalized size = 1.83 \[ -\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}-\frac {\text {Li}_2\left (\frac {a+b x}{a}\right )}{b}+\frac {\log ^2\left (\frac {a}{a+b x}\right )}{2 b}+\frac {\log \left (-\frac {b x}{a}\right ) \log \left (\frac {a}{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {c x}{b x + a}\right )}{b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {c x}{b x + a}\right )}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 97, normalized size = 2.11 \[ -\frac {\ln \left (-\frac {\left (-\frac {a c}{\left (b x +a \right ) b}+\frac {c}{b}\right ) b -c}{c}\right ) \ln \left (-\frac {a c}{\left (b x +a \right ) b}+\frac {c}{b}\right )}{b}-\frac {\dilog \left (-\frac {\left (-\frac {a c}{\left (b x +a \right ) b}+\frac {c}{b}\right ) b -c}{c}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 95, normalized size = 2.07 \[ \frac {\log \left (b x + a\right ) \log \left (\frac {c x}{b x + a}\right )}{b} - \frac {\frac {c \log \left (b x + a\right )^{2}}{b} - \frac {2 \, {\left (\log \left (\frac {b x}{a} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x}{a}\right )\right )} c}{b}}{2 \, c} + \frac {{\left (c \log \left (b x + a\right ) - c \log \relax (x)\right )} \log \left (b x + a\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (\frac {c\,x}{a+b\,x}\right )}{a+b\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (\frac {c x}{a + b x} \right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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