Integrand size = 27, antiderivative size = 24 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {e (f+g x)}{e f-d g}\right )}{g} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2440, 2438} \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {e (f+g x)}{e f-d g}\right )}{g} \]
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Rule 2438
Rule 2440
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {e x}{e f-d g}\right )}{x} \, dx,x,f+g x\right )}{g} \\ & = -\frac {\text {Li}_2\left (\frac {e (f+g x)}{e f-d g}\right )}{g} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {e (f+g x)}{e f-d g}\right )}{g} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(24)=48\).
Time = 1.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25
method | result | size |
derivativedivides | \(\frac {\left (-d g +e f \right ) \operatorname {dilog}\left (-\frac {g e x}{-d g +e f}-\frac {d g}{-d g +e f}\right )}{g \left (d g -e f \right )}\) | \(54\) |
default | \(\frac {\left (-d g +e f \right ) \operatorname {dilog}\left (-\frac {g e x}{-d g +e f}-\frac {d g}{-d g +e f}\right )}{g \left (d g -e f \right )}\) | \(54\) |
risch | \(-\frac {\operatorname {dilog}\left (-\frac {g e x}{-d g +e f}-\frac {d g}{-d g +e f}\right ) d}{d g -e f}+\frac {\operatorname {dilog}\left (-\frac {g e x}{-d g +e f}-\frac {d g}{-d g +e f}\right ) e f}{g \left (d g -e f \right )}\) | \(93\) |
parts | \(\frac {\ln \left (-\frac {g \left (e x +d \right )}{-d g +e f}\right ) \ln \left (g x +f \right )}{g}-\frac {e \left (\frac {\operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{e}+\frac {\ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{e}\right )}{g}\) | \(106\) |
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {{\rm Li}_2\left (\frac {e g x + d g}{e f - d g} + 1\right )}{g} \]
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\[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=\int \frac {\log {\left (- \frac {d g}{- d g + e f} - \frac {e g x}{- d g + e f} \right )}}{f + g x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (23) = 46\).
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.25 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {\log \left (e x + d\right ) \log \left (g x + f\right )}{g} + \frac {\log \left (g x + f\right ) \log \left (-\frac {{\left (e x + d\right )} g}{e f - d g}\right )}{g} + \frac {\log \left (e x + d\right ) \log \left (\frac {e g x + d g}{e f - d g} + 1\right ) + {\rm Li}_2\left (-\frac {e g x + d g}{e f - d g}\right )}{g} \]
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\[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=\int { \frac {\log \left (-\frac {{\left (e x + d\right )} g}{e f - d g}\right )}{g x + f} \,d x } \]
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Time = 1.51 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {g\,\left (d+e\,x\right )}{d\,g-e\,f}\right )}{g} \]
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