Optimal. Leaf size=26 \[ \text {Li}_3(e x) \left (a+b \log \left (c x^n\right )\right )-b n \text {Li}_4(e x) \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2383, 6589} \[ \text {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-b n \text {PolyLog}(4,e x) \]
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 ) \text {Li}_2(e x)}{x} \, dx &=\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)-(b n) \int \frac {\text {Li}_3(e x)}{x} \, dx\\ &=\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)-b n \text {Li}_4(e x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.15 \[ a \text {Li}_3(e x)+b \text {Li}_3(e x) \log \left (c x^n\right )-b n \text {Li}_4(e x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b {\rm Li}_2\left (e x\right ) \log \left (c x^{n}\right ) + a {\rm Li}_2\left (e x\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_2\left (e x\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \polylog \left (2, e x \right )}{x}\, dx \]
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 \[ -\frac {1}{2} \, {\left (b n \log \relax (x)^{2} - 2 \, b \log \relax (x) \log \left (x^{n}\right ) - 2 \, {\left (b \log \relax (c) + a\right )} \log \relax (x)\right )} {\rm Li}_2\left (e x\right ) + \frac {1}{2} \, \int \frac {2 \, b \log \left (-e x + 1\right ) \log \relax (x) \log \left (x^{n}\right ) - {\left (b n \log \relax (x)^{2} - 2 \, {\left (b \log \relax (c) + a\right )} \log \relax (x)\right )} \log \left (-e x + 1\right )}{x}\,{d x} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,e\,x\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.63, size = 26, normalized size = 1.00 \[ a \operatorname {Li}_{3}\left (e x\right ) + b \left (- n \operatorname {Li}_{4}\left (e x\right ) + \log {\left (c x^{n} \right )} \operatorname {Li}_{3}\left (e x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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