Optimal. Leaf size=174 \[ 3 b e n \log (x)-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-3 b e n \log (1-e x)+\frac {3 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \text {Li}_2(e x)-\frac {2 b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-\frac {b n \text {Li}_3(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x} \]
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Rubi [A]
time = 0.11, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {2432, 2442,
36, 29, 31, 2423, 2338, 2438, 6726} \begin {gather*} -\frac {\text {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\text {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-b e n \text {PolyLog}(2,e x)-\frac {2 b n \text {PolyLog}(2,e x)}{x}-\frac {b n \text {PolyLog}(3,e x)}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} b e n \log ^2(x)+3 b e n \log (x)-3 b e n \log (1-e x)+\frac {3 b n \log (1-e x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2338
Rule 2423
Rule 2432
Rule 2438
Rule 2442
Rule 6726
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x^2} \, dx &=-\frac {b n \text {Li}_3(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x}+(b n) \int \frac {\text {Li}_2(e x)}{x^2} \, dx+\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x^2} \, dx\\ &=-\frac {2 b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-\frac {b n \text {Li}_3(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x}-2 \left ((b n) \int \frac {\log (1-e x)}{x^2} \, dx\right )-\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x^2} \, dx\\ &=e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-\frac {2 b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-\frac {b n \text {Li}_3(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x}+(b n) \int \left (-\frac {e \log (x)}{x}-\frac {\log (1-e x)}{x^2}+\frac {e \log (1-e x)}{x}\right ) \, dx-2 \left (-\frac {b n \log (1-e x)}{x}-(b e n) \int \frac {1}{x (1-e x)} \, dx\right )\\ &=e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-\frac {2 b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-\frac {b n \text {Li}_3(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x}-(b n) \int \frac {\log (1-e x)}{x^2} \, dx-(b e n) \int \frac {\log (x)}{x} \, dx+(b e n) \int \frac {\log (1-e x)}{x} \, dx-2 \left (-\frac {b n \log (1-e x)}{x}-(b e n) \int \frac {1}{x} \, dx-\left (b e^2 n\right ) \int \frac {1}{1-e x} \, dx\right )\\ &=-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-2 \left (-b e n \log (x)+b e n \log (1-e x)-\frac {b n \log (1-e x)}{x}\right )-b e n \text {Li}_2(e x)-\frac {2 b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-\frac {b n \text {Li}_3(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x}+(b e n) \int \frac {1}{x (1-e x)} \, dx\\ &=-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-2 \left (-b e n \log (x)+b e n \log (1-e x)-\frac {b n \log (1-e x)}{x}\right )-b e n \text {Li}_2(e x)-\frac {2 b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-\frac {b n \text {Li}_3(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x}+(b e n) \int \frac {1}{x} \, dx+\left (b e^2 n\right ) \int \frac {1}{1-e x} \, dx\\ &=b e n \log (x)-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-b e n \log (1-e x)+\frac {b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-2 \left (-b e n \log (x)+b e n \log (1-e x)-\frac {b n \log (1-e x)}{x}\right )-b e n \text {Li}_2(e x)-\frac {2 b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-\frac {b n \text {Li}_3(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x}\\ \end {align*}
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Mathematica [F]
time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, e x \right )}{x^{2}}\, dx\]
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 [A]
time = 0.41, size = 167, normalized size = 0.96 \begin {gather*} \frac {b n x e \log \left (x\right )^{2} - 2 \, {\left (b n x e + 2 \, b n + a\right )} {\rm Li}_2\left (x e\right ) - 2 \, {\left ({\left (3 \, b n + a\right )} x e - 3 \, b n - a\right )} \log \left (-x e + 1\right ) - 2 \, {\left (b {\rm Li}_2\left (x e\right ) + {\left (b x e - b\right )} \log \left (-x e + 1\right )\right )} \log \left (c\right ) + 2 \, {\left (b x e \log \left (c\right ) - b n {\rm Li}_2\left (x e\right ) + {\left (3 \, b n + a\right )} x e - {\left (b n x e - b n\right )} \log \left (-x e + 1\right )\right )} \log \left (x\right ) - 2 \, {\left (b n \log \left (x\right ) + b n + b \log \left (c\right ) + a\right )} {\rm polylog}\left (3, x e\right )}{2 \, x} \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*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x\right )}{x^{2}}\, dx \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(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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