Optimal. Leaf size=174 \[ -\frac{\text{PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\text{PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-b e n \text{PolyLog}(2,e x)-\frac{2 b n \text{PolyLog}(2,e x)}{x}-\frac{b n \text{PolyLog}(3,e x)}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac{\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{1}{2} b e n \log ^2(x)+3 b e n \log (x)-3 b e n \log (1-e x)+\frac{3 b n \log (1-e x)}{x} \]
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Rubi [A] time = 0.155663, antiderivative size = 174, normalized size of antiderivative = 1., 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 2385
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 2376
Rule 2301
Rule 2391
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*}
Mathematica [F] time = 0.127909, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}(3,e x)}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.238, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\it polylog} \left ( 3,ex \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (e \log \left (x\right ) - \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 \left (c\right ) + \log \left (x^{n}\right )\right )}{\rm Li}_2\left (e x\right ) -{\left (e n x \log \left (x\right ) + 3 \, n + \log \left (c\right )\right )} \log \left (-e x + 1\right ) -{\left (e x \log \left (x\right ) -{\left (e x - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right ) +{\left (n + \log \left (c\right ) + \log \left (x^{n}\right )\right )}{\rm Li}_{3}(e x)}{x} + \int \frac{3 \, e n + e \log \left (c\right ) +{\left (2 \, e^{2} n x - e n\right )} \log \left (x\right )}{e x^{2} - x}\,{d x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.871874, size = 463, normalized size = 2.66 \begin{align*} \frac{b e n x \log \left (x\right )^{2} - 2 \,{\left (b e x - b\right )} \log \left (-e x + 1\right ) \log \left (c\right ) - 2 \,{\left (b e n x + b n \log \left (x\right ) + 2 \, b n + b \log \left (c\right ) + a\right )}{\rm \%iint}\left (e, x, -\frac{\log \left (-e x + 1\right )}{e}, -\frac{\log \left (-e x + 1\right )}{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 e x \log \left (c\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 \left (x\right ) - 2 \,{\left (b n \log \left (x\right ) + b n + b \log \left (c\right ) + a\right )}{\rm polylog}\left (3, e x\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right ) \operatorname{Li}_{3}\left (e x\right )}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}{\rm Li}_{3}(e x)}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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