Optimal. Leaf size=160 \[ -\frac {\text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}+\frac {2 n \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {2 n^2 \text {Li}_4\left (1-\frac {b c-a d}{b (c+d x)}\right )}{g (b c-a d)} \]
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Rubi [A] time = 0.25, 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 = {2506, 2508, 6610} \[ -\frac {\text {PolyLog}\left (2,1-\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}+\frac {2 n \text {PolyLog}\left (3,1-\frac {b c-a d}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {2 n^2 \text {PolyLog}\left (4,1-\frac {b c-a d}{b (c+d x)}\right )}{g (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2506
Rule 2508
Rule 6610
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] time = 0.45, size = 559, normalized size = 3.49 \[ \frac {\log \left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right ) \left (3 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-3 n \log \left (\frac {a+b x}{c+d x}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )\right )+3 n \left (-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )+2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {a+b x}{c+d x}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )\right ) \left (n \log \left (\frac {a+b x}{c+d x}\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {3}{2} \left (2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )-2 \log \left (\frac {a}{b}+x\right ) \log (c+d x)+2 \log (c+d x) \log \left (\frac {a+b x}{c+d x}\right )+2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )-\log ^2\left (\frac {c}{d}+x\right )+2 \log (c+d x) \log \left (\frac {c}{d}+x\right )\right ) \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2-\left (n^2 \left (6 \text {Li}_4\left (\frac {d (a+b x)}{b (c+d x)}\right )+3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )-6 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {a+b x}{c+d x}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^3\left (\frac {a+b x}{c+d x}\right )\right )\right )}{3 g (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} \log \left (\frac {b c - a d}{b d x + b c}\right )}{b d g x^{2} + a c g + {\left (b c + a d\right )} g x}, x\right ) \]
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 \[ \int \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} \log \left (\frac {b c - a d}{{\left (d x + c\right )} b}\right )}{{\left (b g x + a g\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 10.22, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} \ln \left (\frac {-a d +b c}{\left (d x +c \right ) b}\right )}{\left (d x +c \right ) \left (b g x +a g \right )}\, dx \]
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 \[ \int \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} \log \left (\frac {b c - a d}{{\left (d x + c\right )} b}\right )}{{\left (b g x + a g\right )} {\left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (-\frac {a\,d-b\,c}{b\,\left (c+d\,x\right )}\right )\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{\left (a\,g+b\,g\,x\right )\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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