Optimal. Leaf size=104 \[ \frac{6 b^2 n^2 \text{PolyLog}\left (k+3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{q^3}-\frac{3 b n \text{PolyLog}\left (k+2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )^2}{q^2}+\frac{\text{PolyLog}\left (k+1,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )^3}{q}-\frac{6 b^3 n^3 \text{PolyLog}\left (k+4,e x^q\right )}{q^4} \]
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Rubi [A] time = 0.112095, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2383, 6589} \[ \frac{6 b^2 n^2 \text{PolyLog}\left (k+3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{q^3}-\frac{3 b n \text{PolyLog}\left (k+2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )^2}{q^2}+\frac{\text{PolyLog}\left (k+1,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )^3}{q}-\frac{6 b^3 n^3 \text{PolyLog}\left (k+4,e x^q\right )}{q^4} \]
Antiderivative was successfully verified.
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Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_k\left (e x^q\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_{1+k}\left (e x^q\right )}{q}-\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_{1+k}\left (e x^q\right )}{x} \, dx}{q}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_{1+k}\left (e x^q\right )}{q}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_{2+k}\left (e x^q\right )}{q^2}+\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_{2+k}\left (e x^q\right )}{x} \, dx}{q^2}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_{1+k}\left (e x^q\right )}{q}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_{2+k}\left (e x^q\right )}{q^2}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_{3+k}\left (e x^q\right )}{q^3}-\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_{3+k}\left (e x^q\right )}{x} \, dx}{q^3}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \text{Li}_{1+k}\left (e x^q\right )}{q}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_{2+k}\left (e x^q\right )}{q^2}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_{3+k}\left (e x^q\right )}{q^3}-\frac{6 b^3 n^3 \text{Li}_{4+k}\left (e x^q\right )}{q^4}\\ \end{align*}
Mathematica [A] time = 0.0485369, size = 99, normalized size = 0.95 \[ \frac{q^3 \text{PolyLog}\left (k+1,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )^3-3 b n \left (q^2 \text{PolyLog}\left (k+2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (b n \text{PolyLog}\left (k+4,e x^q\right )-q \text{PolyLog}\left (k+3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{q^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}{\it polylog} \left ( k,e{x}^{q} \right ) }{x}}\, 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*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}{\rm Li}_{k}(e x^{q})}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )}{\rm polylog}\left (k, e x^{q}\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{3}{\rm Li}_{k}(e x^{q})}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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