Optimal. Leaf size=142 \[ -\frac {\text {Li}_2(e x) \left (a+b \log \left (c x^n\right )\right )}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-b e n \text {Li}_2(e x)-\frac {b n \text {Li}_2(e x)}{x}-\frac {1}{2} b e n \log ^2(x)+2 b e n \log (x)-2 b e n \log (1-e x)+\frac {2 b n \log (1-e x)}{x} \]
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Rubi [A] time = 0.11, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2385, 2395, 36, 29, 31, 2376, 2301, 2391} \[ -\frac {\text {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-b e n \text {PolyLog}(2,e x)-\frac {b n \text {PolyLog}(2,e x)}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} b e n \log ^2(x)+2 b e n \log (x)-2 b e n \log (1-e x)+\frac {2 b n \log (1-e x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2301
Rule 2376
Rule 2385
Rule 2391
Rule 2395
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x^2} \, dx &=-\frac {b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-(b n) \int \frac {\log (1-e x)}{x^2} \, dx-\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x^2} \, dx\\ &=e \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-\frac {b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}+(b n) \int \left (-\frac {e \log (x)}{x}-\frac {\log (1-e x)}{x^2}+\frac {e \log (1-e x)}{x}\right ) \, dx+(b e n) \int \frac {1}{x (1-e x)} \, dx\\ &=e \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-\frac {b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}-(b n) \int \frac {\log (1-e x)}{x^2} \, dx+(b e n) \int \frac {1}{x} \, dx-(b e n) \int \frac {\log (x)}{x} \, dx+(b e n) \int \frac {\log (1-e x)}{x} \, dx+\left (b e^2 n\right ) \int \frac {1}{1-e x} \, dx\\ &=b e n \log (x)-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-b e n \log (1-e x)+\frac {2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \text {Li}_2(e x)-\frac {b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}+(b e n) \int \frac {1}{x (1-e x)} \, dx\\ &=b e n \log (x)-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-b e n \log (1-e x)+\frac {2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \text {Li}_2(e x)-\frac {b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}+(b e n) \int \frac {1}{x} \, dx+\left (b e^2 n\right ) \int \frac {1}{1-e x} \, dx\\ &=2 b e n \log (x)-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-2 b e n \log (1-e x)+\frac {2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \text {Li}_2(e x)-\frac {b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 115, normalized size = 0.81 \[ \frac {(-\text {Li}_2(e x)+e x \log (x)+(1-e x) \log (1-e x)) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x}+\frac {b n \left (-2 \text {Li}_2(e x) (e x+\log (x)+1)+e x \log ^2(x)+\log (x) (4 e x+(2-2 e x) \log (1-e x))-4 (e x-1) \log (1-e x)\right )}{2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 134, normalized size = 0.94 \[ \frac {b e n x \log \relax (x)^{2} - 2 \, {\left (b e n x + b n + a\right )} {\rm Li}_2\left (e x\right ) + 2 \, {\left (2 \, b n - {\left (2 \, b e n + a e\right )} x + a\right )} \log \left (-e x + 1\right ) - 2 \, {\left (b {\rm Li}_2\left (e x\right ) + {\left (b e x - b\right )} \log \left (-e x + 1\right )\right )} \log \relax (c) + 2 \, {\left (b e x \log \relax (c) - b n {\rm Li}_2\left (e x\right ) + {\left (2 \, b e n + a e\right )} x - {\left (b e n x - b n\right )} \log \left (-e x + 1\right )\right )} \log \relax (x)}{2 \, x} \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_2\left (e x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \polylog \left (2, e x \right )}{x^{2}}\, dx \]
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 \[ {\left (e \log \relax (x) - \frac {{\left (e x - 1\right )} \log \left (-e x + 1\right ) + {\rm Li}_2\left (e x\right )}{x}\right )} a - b {\left (\frac {{\left (n + \log \relax (c) + \log \left (x^{n}\right )\right )} {\rm Li}_2\left (e x\right ) - {\left (e n x \log \relax (x) + 2 \, n + \log \relax (c)\right )} \log \left (-e x + 1\right ) - {\left (e x \log \relax (x) - {\left (e x - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right )}{x} + \int \frac {2 \, e n + e \log \relax (c) + {\left (2 \, e^{2} n x - e n\right )} \log \relax (x)}{e x^{2} - x}\,{d x}\right )} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,e\,x\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{2}\left (e x\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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