Optimal. Leaf size=160 \[ -\frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}+\frac {2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}-\frac {2 n^2 \text {Li}_4\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g} \]
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Rubi [A]
time = 0.16, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {2588, 2590,
6745} \begin {gather*} -\frac {\text {PolyLog}\left (2,1-\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}+\frac {2 n \text {PolyLog}\left (3,1-\frac {b c-a d}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {2 n^2 \text {PolyLog}\left (4,1-\frac {b c-a d}{b (c+d x)}\right )}{g (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2588
Rule 2590
Rule 6745
Rubi steps
\begin {align*} \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{(c+d x) (a g+b g x)} \, dx &=-\frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}+\frac {(2 n) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{g}\\ &=-\frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}+\frac {2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}-\frac {\left (2 n^2\right ) \int \frac {\text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{g}\\ &=-\frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}+\frac {2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}-\frac {2 n^2 \text {Li}_4\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(559\) vs. \(2(160)=320\).
time = 0.32, size = 559, normalized size = 3.49 \begin {gather*} \frac {\log \left (\frac {a+b x}{c+d x}\right ) \left (3 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {a+b x}{c+d x}\right )+n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+\frac {3}{2} \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2 \left (-\log ^2\left (\frac {c}{d}+x\right )-2 \log \left (\frac {a}{b}+x\right ) \log (c+d x)+2 \log \left (\frac {c}{d}+x\right ) \log (c+d x)+2 \log \left (\frac {a+b x}{c+d x}\right ) \log (c+d x)+2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+3 n \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log \left (\frac {a+b x}{c+d x}\right )\right ) \left (\log ^2\left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 \log \left (\frac {a+b x}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )-n^2 \left (\log ^3\left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+3 \log ^2\left (\frac {a+b x}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-6 \log \left (\frac {a+b x}{c+d x}\right ) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )+6 \text {Li}_4\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )}{3 (b c-a d) g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.27, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} \ln \left (\frac {-a d +c b}{b \left (d x +c \right )}\right )}{\left (d x +c \right ) \left (b g x +a g \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2} \log {\left (- \frac {a d}{b c + b d x} + \frac {b c}{b c + b d x} \right )}}{a c + a d x + b c x + b d x^{2}}\, dx}{g} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (-\frac {a\,d-b\,c}{b\,\left (c+d\,x\right )}\right )\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{\left (a\,g+b\,g\,x\right )\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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