Optimal. Leaf size=131 \[ -x \text{PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \text{PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+2 b n x \text{PolyLog}(2,e x)-b n x \text{PolyLog}(3,e x)+\frac{b n \text{PolyLog}(2,e x)}{e}+\frac{(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )-\frac{3 b n (1-e x) \log (1-e x)}{e}-4 b n x \]
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Rubi [A] time = 0.127397, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {2381, 2389, 2295, 2370, 2411, 43, 2351, 2315, 6586} \[ -x \text{PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \text{PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+2 b n x \text{PolyLog}(2,e x)-b n x \text{PolyLog}(3,e x)+\frac{b n \text{PolyLog}(2,e x)}{e}+\frac{(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )-\frac{3 b n (1-e x) \log (1-e x)}{e}-4 b n x \]
Antiderivative was successfully verified.
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Rule 2381
Rule 2389
Rule 2295
Rule 2370
Rule 2411
Rule 43
Rule 2351
Rule 2315
Rule 6586
Rubi steps
\begin{align*} \int \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x) \, dx &=-b n x \text{Li}_3(e x)+x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)+(b n) \int \text{Li}_2(e x) \, dx-\int \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x) \, dx\\ &=2 b n x \text{Li}_2(e x)-x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)-b n x \text{Li}_3(e x)+x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)+2 ((b n) \int \log (1-e x) \, dx)-\int \left (a+b \log \left (c x^n\right )\right ) \log (1-e x) \, dx\\ &=x \left (a+b \log \left (c x^n\right )\right )+\frac{(1-e x) \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{e}+2 b n x \text{Li}_2(e x)-x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)-b n x \text{Li}_3(e x)+x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)+(b n) \int \left (-1-\frac{(1-e x) \log (1-e x)}{e x}\right ) \, dx-2 \frac{(b n) \operatorname{Subst}(\int \log (x) \, dx,x,1-e x)}{e}\\ &=-b n x+x \left (a+b \log \left (c x^n\right )\right )+\frac{(1-e x) \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{e}-2 \left (b n x+\frac{b n (1-e x) \log (1-e x)}{e}\right )+2 b n x \text{Li}_2(e x)-x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)-b n x \text{Li}_3(e x)+x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)-\frac{(b n) \int \frac{(1-e x) \log (1-e x)}{x} \, dx}{e}\\ &=-b n x+x \left (a+b \log \left (c x^n\right )\right )+\frac{(1-e x) \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{e}-2 \left (b n x+\frac{b n (1-e x) \log (1-e x)}{e}\right )+2 b n x \text{Li}_2(e x)-x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)-b n x \text{Li}_3(e x)+x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)+\frac{(b n) \operatorname{Subst}\left (\int \frac{x \log (x)}{\frac{1}{e}-\frac{x}{e}} \, dx,x,1-e x\right )}{e^2}\\ &=-b n x+x \left (a+b \log \left (c x^n\right )\right )+\frac{(1-e x) \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{e}-2 \left (b n x+\frac{b n (1-e x) \log (1-e x)}{e}\right )+2 b n x \text{Li}_2(e x)-x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)-b n x \text{Li}_3(e x)+x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)+\frac{(b n) \operatorname{Subst}\left (\int \left (-e \log (x)-\frac{e \log (x)}{-1+x}\right ) \, dx,x,1-e x\right )}{e^2}\\ &=-b n x+x \left (a+b \log \left (c x^n\right )\right )+\frac{(1-e x) \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{e}-2 \left (b n x+\frac{b n (1-e x) \log (1-e x)}{e}\right )+2 b n x \text{Li}_2(e x)-x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)-b n x \text{Li}_3(e x)+x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)-\frac{(b n) \operatorname{Subst}(\int \log (x) \, dx,x,1-e x)}{e}-\frac{(b n) \operatorname{Subst}\left (\int \frac{\log (x)}{-1+x} \, dx,x,1-e x\right )}{e}\\ &=-2 b n x+x \left (a+b \log \left (c x^n\right )\right )-\frac{b n (1-e x) \log (1-e x)}{e}+\frac{(1-e x) \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{e}-2 \left (b n x+\frac{b n (1-e x) \log (1-e x)}{e}\right )+\frac{b n \text{Li}_2(e x)}{e}+2 b n x \text{Li}_2(e x)-x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)-b n x \text{Li}_3(e x)+x \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)\\ \end{align*}
Mathematica [F] time = 0.0799283, size = 0, normalized size = 0. \[ \int \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}(3,e x) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.273, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\it polylog} \left ( 3,ex \right ) \, 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*} -b{\left (\frac{{\left (e x \log \left (x^{n}\right ) -{\left (2 \, e n - e \log \left (c\right )\right )} x\right )}{\rm Li}_2\left (e x\right ) -{\left ({\left (3 \, e n - e \log \left (c\right )\right )} x - n \log \left (x\right )\right )} \log \left (-e x + 1\right ) -{\left (e x -{\left (e x - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right ) -{\left (e x \log \left (x^{n}\right ) -{\left (e n - e \log \left (c\right )\right )} x\right )}{\rm Li}_{3}(e x)}{e} - \int -\frac{{\left (4 \, e n - e \log \left (c\right )\right )} x - n \log \left (x\right ) - n}{e x - 1}\,{d x}\right )} - \frac{{\left (e x{\rm Li}_2\left (e x\right ) - e x{\rm Li}_{3}(e x) - e x +{\left (e x - 1\right )} \log \left (-e x + 1\right )\right )} a}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.98276, size = 464, normalized size = 3.54 \begin{align*} -\frac{{\left (4 \, b e n - a e\right )} x +{\left (b e n x \log \left (x\right ) + b e x \log \left (c\right ) - b n -{\left (2 \, b e n - a e\right )} x\right )}{\rm \%iint}\left (e, x, -\frac{\log \left (-e x + 1\right )}{e}, -\frac{\log \left (-e x + 1\right )}{x}\right ) +{\left (3 \, b n -{\left (3 \, b e n - a e\right )} x - a\right )} \log \left (-e x + 1\right ) -{\left (b e x -{\left (b e x - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) -{\left (b e n x -{\left (b e n x - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right ) -{\left (b e n x \log \left (x\right ) + b e x \log \left (c\right ) -{\left (b e n - a e\right )} x\right )}{\rm polylog}\left (3, e x\right )}{e} \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 \left (a + b \log{\left (c x^{n} \right )}\right ) \operatorname{Li}_{3}\left (e x\right )\, 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{\left (b \log \left (c x^{n}\right ) + a\right )}{\rm Li}_{3}(e x)\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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