Optimal. Leaf size=174 \[ -\frac {\text {Li}_2(e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\text {Li}_3(e x) \left (a+b \log \left (c x^n\right )\right )}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-b e n \text {Li}_2(e x)-\frac {2 b n \text {Li}_2(e x)}{x}-\frac {b n \text {Li}_3(e x)}{x}-\frac {1}{2} b e n \log ^2(x)+3 b e n \log (x)-3 b e n \log (1-e x)+\frac {3 b n \log (1-e x)}{x} \]
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Rubi [A] time = 0.16, 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 = {2385, 2395, 36, 29, 31, 2376, 2301, 2391, 6591} \[ -\frac {\text {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\text {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-b e n \text {PolyLog}(2,e x)-\frac {2 b n \text {PolyLog}(2,e x)}{x}-\frac {b n \text {PolyLog}(3,e x)}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} b e n \log ^2(x)+3 b e n \log (x)-3 b e n \log (1-e x)+\frac {3 b n \log (1-e x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2301
Rule 2376
Rule 2385
Rule 2391
Rule 2395
Rule 6591
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 = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.62, size = 156, normalized size = 0.90 \[ \frac {b e n x \log \relax (x)^{2} - 2 \, {\left (b e n x + 2 \, b n + a\right )} {\rm Li}_2\left (e x\right ) + 2 \, {\left (3 \, b n - {\left (3 \, b e n + a e\right )} x + a\right )} \log \left (-e x + 1\right ) - 2 \, {\left (b {\rm Li}_2\left (e x\right ) + {\left (b e x - b\right )} \log \left (-e x + 1\right )\right )} \log \relax (c) + 2 \, {\left (b e x \log \relax (c) - b n {\rm Li}_2\left (e x\right ) + {\left (3 \, b e n + a e\right )} x - {\left (b e n x - b n\right )} \log \left (-e x + 1\right )\right )} \log \relax (x) - 2 \, {\left (b n \log \relax (x) + b n + b \log \relax (c) + a\right )} {\rm polylog}\left (3, e x\right )}{2 \, x} \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x)}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \polylog \left (3, e x \right )}{x^{2}}\, dx \]
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 \[ {\left (e \log \relax (x) - \frac {{\left (e x - 1\right )} \log \left (-e x + 1\right ) + {\rm Li}_2\left (e x\right ) + {\rm Li}_{3}(e x)}{x}\right )} a - b {\left (\frac {{\left (2 \, n + \log \relax (c) + \log \left (x^{n}\right )\right )} {\rm Li}_2\left (e x\right ) - {\left (e n x \log \relax (x) + 3 \, n + \log \relax (c)\right )} \log \left (-e x + 1\right ) - {\left (e x \log \relax (x) - {\left (e x - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right ) + {\left (n + \log \relax (c) + \log \left (x^{n}\right )\right )} {\rm Li}_{3}(e x)}{x} + \int \frac {3 \, e n + e \log \relax (c) + {\left (2 \, e^{2} n x - e n\right )} \log \relax (x)}{e x^{2} - x}\,{d x}\right )} \]
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 \[ \text {Hanged} \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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