Integrand size = 19, antiderivative size = 46 \[ \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx=-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {a}{a+b x}\right )}{b} \]
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Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2544, 2458, 2378, 2370, 2352} \[ \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx=-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {a}{a+b x}\right )}{b} \]
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Rule 2352
Rule 2370
Rule 2378
Rule 2458
Rule 2544
Rubi steps \begin{align*} \text {integral}& = -\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 \text {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 \text {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 \text {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*}
Time = 0.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.83 \[ \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx=\frac {\log \left (-\frac {b x}{a}\right ) \log \left (\frac {a}{a+b x}\right )}{b}+\frac {\log ^2\left (\frac {a}{a+b x}\right )}{2 b}-\frac {\log \left (\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (2,\frac {a+b x}{a}\right )}{b} \]
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Time = 1.90 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.50
method | result | size |
parts | \(\frac {\ln \left (\frac {c x}{b x +a}\right ) \ln \left (b x +a \right )}{b}-\frac {-\frac {c \ln \left (b x +a \right )^{2}}{2}+c \left (\operatorname {dilog}\left (-\frac {x b}{a}\right )+\ln \left (b x +a \right ) \ln \left (-\frac {x b}{a}\right )\right )}{b c}\) | \(69\) |
derivativedivides | \(-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) b -c}{c}\right )}{b}-\frac {\ln \left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) \ln \left (-\frac {\left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) b -c}{c}\right )}{b}\) | \(97\) |
default | \(-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) b -c}{c}\right )}{b}-\frac {\ln \left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) \ln \left (-\frac {\left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) b -c}{c}\right )}{b}\) | \(97\) |
risch | \(-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) b -c}{c}\right )}{b}-\frac {\ln \left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) \ln \left (-\frac {\left (\frac {c}{b}-\frac {a c}{b \left (b x +a \right )}\right ) b -c}{c}\right )}{b}\) | \(97\) |
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\[ \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx=\int { \frac {\log \left (\frac {c x}{b x + a}\right )}{b x + a} \,d x } \]
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\[ \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx=\int \frac {\log {\left (\frac {c x}{a + b x} \right )}}{a + b x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (45) = 90\).
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.07 \[ \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx=\frac {\log \left (b x + a\right ) \log \left (\frac {c x}{b x + a}\right )}{b} - \frac {\frac {c \log \left (b x + a\right )^{2}}{b} - \frac {2 \, {\left (\log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a}\right )\right )} c}{b}}{2 \, c} + \frac {{\left (c \log \left (b x + a\right ) - c \log \left (x\right )\right )} \log \left (b x + a\right )}{b c} \]
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\[ \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx=\int { \frac {\log \left (\frac {c x}{b x + a}\right )}{b x + a} \,d x } \]
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Timed out. \[ \int \frac {\log \left (\frac {c x}{a+b x}\right )}{a+b x} \, dx=\int \frac {\ln \left (\frac {c\,x}{a+b\,x}\right )}{a+b\,x} \,d x \]
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