Optimal. Leaf size=102 \[ \frac{q^2 \text{Unintegrable}\left (\frac{\text{PolyLog}\left (k-2,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )},x\right )}{2 b^2 n^2}-\frac{q \text{PolyLog}\left (k-1,e x^q\right )}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{\text{PolyLog}\left (k,e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rubi [A] time = 0.111534, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\text{Li}_k\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx &=-\frac{\text{Li}_k\left (e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac{q \int \frac{\text{Li}_{-1+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n}\\ &=-\frac{q \text{Li}_{-1+k}\left (e x^q\right )}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{\text{Li}_k\left (e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac{q^2 \int \frac{\text{Li}_{-2+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b^2 n^2}\\ \end{align*}
Mathematica [A] time = 0.0474568, size = 0, normalized size = 0. \[ \int \frac{\text{PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it polylog} \left ( k,e{x}^{q} \right ) }{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm polylog}\left (k, e x^{q}\right )}{b^{3} x \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b x \log \left (c x^{n}\right ) + a^{3} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{k}\left (e x^{q}\right )}{x \left (a + b \log{\left (c x^{n} \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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