Optimal. Leaf size=26 \[ \frac {\text {Li}_k\left (e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A]
time = 0.07, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 = {2431}
\begin {gather*} \frac {\text {PolyLog}\left (k,e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2431
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*}
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Mathematica [F]
time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {q \polylog \left (k -1, e \,x^{q}\right )}{b n x \left (a +b \ln \left (c \,x^{n}\right )\right )}-\frac {\polylog \left (k , e \,x^{q}\right )}{x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \left (- \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}}\right )\, 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {q\,\mathrm {polylog}\left (k-1,e\,x^q\right )}{b\,n\,x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}-\frac {\mathrm {polylog}\left (k,e\,x^q\right )}{x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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