Optimal. Leaf size=24 \[ -\frac {\text {Li}_2\left (\frac {e (f+g x)}{e f-d g}\right )}{g} \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2393, 2391} \[ -\frac {\text {PolyLog}\left (2,\frac {e (f+g x)}{e f-d g}\right )}{g} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rubi steps
\begin {align*} \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx &=\frac {\operatorname {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*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ -\frac {\text {Li}_2\left (\frac {e (f+g x)}{e f-d g}\right )}{g} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 27, normalized size = 1.12 \[ -\frac {{\rm Li}_2\left (\frac {e g x + d g}{e f - d g} + 1\right )}{g} \]
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 \[ \int \frac {\log \left (-\frac {{\left (e x + d\right )} g}{e f - d g}\right )}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.46 \[ -\frac {\dilog \left (\frac {e g x}{d g -e f}+\frac {d g}{d g -e f}\right )}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.67, size = 102, normalized size = 4.25 \[ -\frac {\log \left (e x + d\right ) \log \left (g x + f\right )}{g} + \frac {\log \left (g x + f\right ) \log \left (-\frac {{\left (e x + d\right )} g}{e f - d g}\right )}{g} + \frac {\log \left (e x + d\right ) \log \left (\frac {e g x + d g}{e f - d g} + 1\right ) + {\rm Li}_2\left (-\frac {e g x + d g}{e f - d g}\right )}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 23, normalized size = 0.96 \[ -\frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {g\,\left (d+e\,x\right )}{d\,g-e\,f}\right )}{g} \]
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 \[ \int \frac {\log {\left (- \frac {d g}{- d g + e f} - \frac {e g x}{- d g + e f} \right )}}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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