Optimal. Leaf size=104 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_{2+k}\left (e x^q\right )}{q^2}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_{3+k}\left (e x^q\right )}{q^3}-\frac {6 b^3 n^3 \text {Li}_{4+k}\left (e x^q\right )}{q^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 104, normalized size of antiderivative = 1.00, 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 = {2430, 6724}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2430
Rule 6724
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*}
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Mathematica [A]
time = 0.04, size = 99, normalized size = 0.95 \begin {gather*} \frac {q^3 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_{1+k}\left (e x^q\right )-3 b n \left (q^2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_{2+k}\left (e x^q\right )+2 b n \left (-q \left (a+b \log \left (c x^n\right )\right ) \text {Li}_{3+k}\left (e x^q\right )+b n \text {Li}_{4+k}\left (e x^q\right )\right )\right )}{q^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \polylog \left (k , e \,x^{q}\right )}{x}\, dx\]
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*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3} \operatorname {Li}_{k}\left (e x^{q}\right )}{x}\, dx \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 = -0.01 \begin {gather*} \int \frac {\mathrm {polylog}\left (k,e\,x^q\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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