Optimal. Leaf size=28 \[ b n \text {Li}_3(-e x)-\text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2374, 6589} \[ b n \text {PolyLog}(3,-e x)-\text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx &=-\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+(b n) \int \frac {\text {Li}_2(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+b n \text {Li}_3(-e x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.21 \[ -a \text {Li}_2(-e x)-b \text {Li}_2(-e x) \log \left (c x^n\right )+b n \text {Li}_3(-e x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a \log \left (e x + 1\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (e x + 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (e x +1\right )}{x}\, dx \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (e x + 1\right )}{x}\,{d x} \]
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 \[ \int \frac {\ln \left (e\,x+1\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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