Optimal. Leaf size=238 \[ \frac {1}{8} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {\text {Li}_2(e x) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {\text {Li}_3(e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{8 x^2}-\frac {1}{8} b e^2 n \text {Li}_2(e x)-\frac {1}{16} b e^2 n \log ^2(x)+\frac {3}{16} b e^2 n \log (x)-\frac {3}{16} b e^2 n \log (1-e x)-\frac {b n \text {Li}_2(e x)}{4 x^2}-\frac {b n \text {Li}_3(e x)}{4 x^2}+\frac {3 b n \log (1-e x)}{16 x^2}-\frac {5 b e n}{16 x} \]
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Rubi [A] time = 0.22, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2385, 2395, 44, 2376, 2301, 2391, 6591} \[ -\frac {\text {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {\text {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{8} b e^2 n \text {PolyLog}(2,e x)-\frac {b n \text {PolyLog}(2,e x)}{4 x^2}-\frac {b n \text {PolyLog}(3,e x)}{4 x^2}+\frac {1}{8} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{8 x^2}-\frac {1}{16} b e^2 n \log ^2(x)+\frac {3}{16} b e^2 n \log (x)-\frac {3}{16} b e^2 n \log (1-e x)+\frac {3 b n \log (1-e x)}{16 x^2}-\frac {5 b e n}{16 x} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2301
Rule 2376
Rule 2385
Rule 2391
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x^3} \, dx &=-\frac {b n \text {Li}_3(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{2 x^2}+\frac {1}{2} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x^3} \, dx+\frac {1}{4} (b n) \int \frac {\text {Li}_2(e x)}{x^3} \, dx\\ &=-\frac {b n \text {Li}_2(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{4 x^2}-\frac {b n \text {Li}_3(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{2 x^2}-\frac {1}{4} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x^3} \, dx-2 \left (\frac {1}{8} (b n) \int \frac {\log (1-e x)}{x^3} \, dx\right )\\ &=-\frac {e \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {1}{8} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} e^2 \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)}{8 x^2}-\frac {b n \text {Li}_2(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{4 x^2}-\frac {b n \text {Li}_3(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{2 x^2}+\frac {1}{4} (b n) \int \left (\frac {e}{2 x^2}-\frac {e^2 \log (x)}{2 x}-\frac {\log (1-e x)}{2 x^3}+\frac {e^2 \log (1-e x)}{2 x}\right ) \, dx-2 \left (-\frac {b n \log (1-e x)}{16 x^2}-\frac {1}{16} (b e n) \int \frac {1}{x^2 (1-e x)} \, dx\right )\\ &=-\frac {b e n}{8 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {1}{8} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} e^2 \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)}{8 x^2}-\frac {b n \text {Li}_2(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{4 x^2}-\frac {b n \text {Li}_3(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{2 x^2}-\frac {1}{8} (b n) \int \frac {\log (1-e x)}{x^3} \, dx-2 \left (-\frac {b n \log (1-e x)}{16 x^2}-\frac {1}{16} (b e n) \int \left (\frac {1}{x^2}+\frac {e}{x}-\frac {e^2}{-1+e x}\right ) \, dx\right )-\frac {1}{8} \left (b e^2 n\right ) \int \frac {\log (x)}{x} \, dx+\frac {1}{8} \left (b e^2 n\right ) \int \frac {\log (1-e x)}{x} \, dx\\ &=-\frac {b e n}{8 x}-\frac {1}{16} b e^2 n \log ^2(x)-\frac {e \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {1}{8} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1-e x)}{16 x^2}-\frac {1}{8} e^2 \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)}{8 x^2}-2 \left (\frac {b e n}{16 x}-\frac {1}{16} b e^2 n \log (x)+\frac {1}{16} b e^2 n \log (1-e x)-\frac {b n \log (1-e x)}{16 x^2}\right )-\frac {1}{8} b e^2 n \text {Li}_2(e x)-\frac {b n \text {Li}_2(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{4 x^2}-\frac {b n \text {Li}_3(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{2 x^2}+\frac {1}{16} (b e n) \int \frac {1}{x^2 (1-e x)} \, dx\\ &=-\frac {b e n}{8 x}-\frac {1}{16} b e^2 n \log ^2(x)-\frac {e \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {1}{8} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1-e x)}{16 x^2}-\frac {1}{8} e^2 \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)}{8 x^2}-2 \left (\frac {b e n}{16 x}-\frac {1}{16} b e^2 n \log (x)+\frac {1}{16} b e^2 n \log (1-e x)-\frac {b n \log (1-e x)}{16 x^2}\right )-\frac {1}{8} b e^2 n \text {Li}_2(e x)-\frac {b n \text {Li}_2(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{4 x^2}-\frac {b n \text {Li}_3(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{2 x^2}+\frac {1}{16} (b e n) \int \left (\frac {1}{x^2}+\frac {e}{x}-\frac {e^2}{-1+e x}\right ) \, dx\\ &=-\frac {3 b e n}{16 x}+\frac {1}{16} b e^2 n \log (x)-\frac {1}{16} b e^2 n \log ^2(x)-\frac {e \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {1}{8} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b e^2 n \log (1-e x)+\frac {b n \log (1-e x)}{16 x^2}-\frac {1}{8} e^2 \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)}{8 x^2}-2 \left (\frac {b e n}{16 x}-\frac {1}{16} b e^2 n \log (x)+\frac {1}{16} b e^2 n \log (1-e x)-\frac {b n \log (1-e x)}{16 x^2}\right )-\frac {1}{8} b e^2 n \text {Li}_2(e x)-\frac {b n \text {Li}_2(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{4 x^2}-\frac {b n \text {Li}_3(e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{2 x^2}\\ \end {align*}
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Mathematica [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.55, size = 221, normalized size = 0.93 \[ \frac {b e^{2} n x^{2} \log \relax (x)^{2} - {\left (5 \, b e n + 2 \, a e\right )} x - 2 \, {\left (b e^{2} n x^{2} + 2 \, b n + 2 \, a\right )} {\rm Li}_2\left (e x\right ) - {\left ({\left (3 \, b e^{2} n + 2 \, a e^{2}\right )} x^{2} - 3 \, b n - 2 \, a\right )} \log \left (-e x + 1\right ) - 2 \, {\left (b e x + 2 \, b {\rm Li}_2\left (e x\right ) + {\left (b e^{2} x^{2} - b\right )} \log \left (-e x + 1\right )\right )} \log \relax (c) + {\left (2 \, b e^{2} x^{2} \log \relax (c) - 2 \, b e n x + {\left (3 \, b e^{2} n + 2 \, a e^{2}\right )} x^{2} - 4 \, b n {\rm Li}_2\left (e x\right ) - 2 \, {\left (b e^{2} n x^{2} - b n\right )} \log \left (-e x + 1\right )\right )} \log \relax (x) - 4 \, {\left (2 \, b n \log \relax (x) + b n + 2 \, b \log \relax (c) + 2 \, a\right )} {\rm polylog}\left (3, e x\right )}{16 \, x^{2}} \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x)}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \polylog \left (3, e x \right )}{x^{3}}\, dx \]
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 \[ \frac {1}{8} \, {\left (e^{2} \log \relax (x) - \frac {e x + {\left (e^{2} x^{2} - 1\right )} \log \left (-e x + 1\right ) + 2 \, {\rm Li}_2\left (e x\right ) + 4 \, {\rm Li}_{3}(e x)}{x^{2}}\right )} a - \frac {1}{16} \, b {\left (\frac {4 \, {\left (n + \log \relax (c) + \log \left (x^{n}\right )\right )} {\rm Li}_2\left (e x\right ) - {\left (2 \, e^{2} n x^{2} \log \relax (x) + 3 \, n + 2 \, \log \relax (c)\right )} \log \left (-e x + 1\right ) - 2 \, {\left (e^{2} x^{2} \log \relax (x) - e x - {\left (e^{2} x^{2} - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right ) + 4 \, {\left (n + 2 \, \log \relax (c) + 2 \, \log \left (x^{n}\right )\right )} {\rm Li}_{3}(e x)}{x^{2}} + 16 \, \int -\frac {2 \, e^{2} n x - 5 \, e n - 2 \, e \log \relax (c) - 2 \, {\left (2 \, e^{3} n x^{2} - e^{2} n x\right )} \log \relax (x)}{16 \, {\left (e x^{3} - x^{2}\right )}}\,{d x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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