Optimal. Leaf size=26 \[ \frac{\text{PolyLog}\left (k,e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.109864, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 57, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.018, Rules used = {2384} \[ \frac{\text{PolyLog}\left (k,e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2384
Rubi steps
\begin{align*} \int \left (\frac{q \text{Li}_{-1+k}\left (e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac{\text{Li}_k\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx &=\frac{q \int \frac{\text{Li}_{-1+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )} \, dx}{b n}-\int \frac{\text{Li}_k\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx\\ &=\frac{\text{Li}_k\left (e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [F] time = 0.147066, size = 0, normalized size = 0. \[ \int \left (\frac{q \text{PolyLog}\left (-1+k,e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac{\text{PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{q{\it polylog} \left ( -1+k,e{x}^{q} \right ) }{bnx \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}-{\frac{{\it polylog} \left ( k,e{x}^{q} \right ) }{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{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*} \int \frac{q{\rm Li}_{k - 1}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n x} - \frac{{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} 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{b n{\rm polylog}\left (k, e x^{q}\right ) -{\left (b q \log \left (c x^{n}\right ) + a q\right )}{\rm polylog}\left (k - 1, e x^{q}\right )}{b^{3} n x \log \left (c x^{n}\right )^{2} + 2 \, a b^{2} n x \log \left (c x^{n}\right ) + a^{2} b n x}, x\right ) \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*} \frac{\int \frac{a q \operatorname{Li}_{k - 1}\left (e x^{q}\right )}{a^{2} x + 2 a b x \log{\left (c x^{n} \right )} + b^{2} x \log{\left (c x^{n} \right )}^{2}}\, dx + \int - \frac{b n \operatorname{Li}_{k}\left (e x^{q}\right )}{a^{2} x + 2 a b x \log{\left (c x^{n} \right )} + b^{2} x \log{\left (c x^{n} \right )}^{2}}\, dx + \int \frac{b q \log{\left (c x^{n} \right )} \operatorname{Li}_{k - 1}\left (e x^{q}\right )}{a^{2} x + 2 a b x \log{\left (c x^{n} \right )} + b^{2} x \log{\left (c x^{n} \right )}^{2}}\, dx}{b n} \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{q{\rm Li}_{k - 1}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n x} - \frac{{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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