Optimal. Leaf size=146 \[ -\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {Li}_2(-e x)}{2 e^2}+\frac {b n \log (e x+1)}{4 e^2}-\frac {1}{4} b n x^2 \log (e x+1)-\frac {3 b n x}{4 e}+\frac {1}{4} b n x^2 \]
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Rubi [A] time = 0.08, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2395, 43, 2376, 2391} \[ -\frac {b n \text {PolyLog}(2,-e x)}{2 e^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (e x+1)}{4 e^2}-\frac {1}{4} b n x^2 \log (e x+1)-\frac {3 b n x}{4 e}+\frac {1}{4} b n x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2376
Rule 2391
Rule 2395
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-(b n) \int \left (\frac {1}{2 e}-\frac {x}{4}-\frac {\log (1+e x)}{2 e^2 x}+\frac {1}{2} x \log (1+e x)\right ) \, dx\\ &=-\frac {b n x}{2 e}+\frac {1}{8} b n x^2+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {1}{2} (b n) \int x \log (1+e x) \, dx+\frac {(b n) \int \frac {\log (1+e x)}{x} \, dx}{2 e^2}\\ &=-\frac {b n x}{2 e}+\frac {1}{8} b n x^2+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \text {Li}_2(-e x)}{2 e^2}+\frac {1}{4} (b e n) \int \frac {x^2}{1+e x} \, dx\\ &=-\frac {b n x}{2 e}+\frac {1}{8} b n x^2+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \text {Li}_2(-e x)}{2 e^2}+\frac {1}{4} (b e n) \int \left (-\frac {1}{e^2}+\frac {x}{e}+\frac {1}{e^2 (1+e x)}\right ) \, dx\\ &=-\frac {3 b n x}{4 e}+\frac {1}{4} b n x^2+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1+e x)}{4 e^2}-\frac {1}{4} b n x^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \text {Li}_2(-e x)}{2 e^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 131, normalized size = 0.90 \[ \frac {-a e^2 x^2+2 a e^2 x^2 \log (e x+1)+2 a e x-2 a \log (e x+1)+b \left (2 \left (e^2 x^2-1\right ) \log (e x+1)+e x (2-e x)\right ) \log \left (c x^n\right )+b e^2 n x^2-b e^2 n x^2 \log (e x+1)-2 b n \text {Li}_2(-e x)-3 b e n x+b n \log (e x+1)}{4 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a x \log \left (e x + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left (e x + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 725, normalized size = 4.97 \[ \frac {3 a}{4 e^{2}}-\frac {a \,x^{2}}{4}+\left (\frac {b \,x^{2} \ln \left (e x +1\right )}{2}-\frac {\left (e^{2} x^{2}-2 e x +2 \ln \left (e x +1\right )\right ) b}{4 e^{2}}\right ) \ln \left (x^{n}\right )-\frac {b n \dilog \left (e x +1\right )}{2 e^{2}}-\frac {b n}{e^{2}}-\frac {b \,x^{2} \ln \relax (c )}{4}+\frac {b n \,x^{2}}{4}+\frac {b x \ln \relax (c )}{2 e}-\frac {b \ln \relax (c ) \ln \left (e x +1\right )}{2 e^{2}}+\frac {3 b \ln \relax (c )}{4 e^{2}}-\frac {a \ln \left (e x +1\right )}{2 e^{2}}+\frac {a \,x^{2} \ln \left (e x +1\right )}{2}+\frac {b \,x^{2} \ln \relax (c ) \ln \left (e x +1\right )}{2}-\frac {3 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8 e^{2}}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e x +1\right )}{4}-\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4 e}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e x +1\right )}{4 e^{2}}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +1\right )}{4}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4 e}+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +1\right )}{4 e^{2}}+\frac {3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8 e^{2}}+\frac {3 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8 e^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +1\right )}{4 e^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +1\right )}{4 e^{2}}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +1\right )}{4}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +1\right )}{4}+\frac {i \pi b \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}+\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 e}+\frac {i \pi b x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4 e}-\frac {3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8 e^{2}}+\frac {a x}{2 e}-\frac {3 b n x}{4 e}+\frac {b n \ln \left (e x +1\right )}{4 e^{2}}-\frac {b n \,x^{2} \ln \left (e x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 178, normalized size = 1.22 \[ -\frac {{\left (\log \left (e x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-e x\right )\right )} b n}{2 \, e^{2}} + \frac {{\left (b {\left (n - 2 \, \log \relax (c)\right )} - 2 \, a\right )} \log \left (e x + 1\right )}{4 \, e^{2}} - \frac {{\left (a e^{2} - {\left (e^{2} n - e^{2} \log \relax (c)\right )} b\right )} x^{2} + {\left ({\left (3 \, e n - 2 \, e \log \relax (c)\right )} b - 2 \, a e\right )} x - {\left ({\left (2 \, a e^{2} - {\left (e^{2} n - 2 \, e^{2} \log \relax (c)\right )} b\right )} x^{2} + 2 \, b n \log \relax (x)\right )} \log \left (e x + 1\right ) + {\left (b e^{2} x^{2} - 2 \, b e x - 2 \, {\left (b e^{2} x^{2} - b\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )}{4 \, e^{2}} \]
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 \[ \int x\,\ln \left (e\,x+1\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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