Optimal. Leaf size=195 \[ -\frac {5 b e n}{36 x^2}+\frac {4 b e^2 n}{9 x}+\frac {1}{9} b e^3 n \log (x)-\frac {1}{6} b e^3 n \log ^2(x)-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x}+\frac {1}{3} e^3 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b e^3 n \log (1+e x)-\frac {b n \log (1+e x)}{9 x^3}-\frac {1}{3} e^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{3 x^3}-\frac {1}{3} b e^3 n \text {Li}_2(-e x) \]
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Rubi [A]
time = 0.08, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2442, 46, 2423,
2338, 2438} \begin {gather*} -\frac {1}{3} b e^3 n \text {PolyLog}(2,-e x)+\frac {1}{3} e^3 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} e^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 x^2}-\frac {1}{6} b e^3 n \log ^2(x)+\frac {1}{9} b e^3 n \log (x)-\frac {1}{9} b e^3 n \log (e x+1)+\frac {4 b e^2 n}{9 x}-\frac {b n \log (e x+1)}{9 x^3}-\frac {5 b e n}{36 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2338
Rule 2423
Rule 2438
Rule 2442
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x^4} \, dx &=-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x}+\frac {1}{3} e^3 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} e^3 \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)}{3 x^3}-(b n) \int \left (-\frac {e}{6 x^3}+\frac {e^2}{3 x^2}+\frac {e^3 \log (x)}{3 x}-\frac {\log (1+e x)}{3 x^4}-\frac {e^3 \log (1+e x)}{3 x}\right ) \, dx\\ &=-\frac {b e n}{12 x^2}+\frac {b e^2 n}{3 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x}+\frac {1}{3} e^3 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} e^3 \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)}{3 x^3}+\frac {1}{3} (b n) \int \frac {\log (1+e x)}{x^4} \, dx-\frac {1}{3} \left (b e^3 n\right ) \int \frac {\log (x)}{x} \, dx+\frac {1}{3} \left (b e^3 n\right ) \int \frac {\log (1+e x)}{x} \, dx\\ &=-\frac {b e n}{12 x^2}+\frac {b e^2 n}{3 x}-\frac {1}{6} b e^3 n \log ^2(x)-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x}+\frac {1}{3} e^3 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log (1+e x)}{9 x^3}-\frac {1}{3} e^3 \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)}{3 x^3}-\frac {1}{3} b e^3 n \text {Li}_2(-e x)+\frac {1}{9} (b e n) \int \frac {1}{x^3 (1+e x)} \, dx\\ &=-\frac {b e n}{12 x^2}+\frac {b e^2 n}{3 x}-\frac {1}{6} b e^3 n \log ^2(x)-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x}+\frac {1}{3} e^3 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log (1+e x)}{9 x^3}-\frac {1}{3} e^3 \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)}{3 x^3}-\frac {1}{3} b e^3 n \text {Li}_2(-e x)+\frac {1}{9} (b e n) \int \left (\frac {1}{x^3}-\frac {e}{x^2}+\frac {e^2}{x}-\frac {e^3}{1+e x}\right ) \, dx\\ &=-\frac {5 b e n}{36 x^2}+\frac {4 b e^2 n}{9 x}+\frac {1}{9} b e^3 n \log (x)-\frac {1}{6} b e^3 n \log ^2(x)-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x}+\frac {1}{3} e^3 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b e^3 n \log (1+e x)-\frac {b n \log (1+e x)}{9 x^3}-\frac {1}{3} e^3 \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)}{3 x^3}-\frac {1}{3} b e^3 n \text {Li}_2(-e x)\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 206, normalized size = 1.06 \begin {gather*} -\frac {6 a e x+5 b e n x-12 a e^2 x^2-16 b e^2 n x^2+6 b e^3 n x^3 \log ^2(x)+6 b e x \log \left (c x^n\right )-12 b e^2 x^2 \log \left (c x^n\right )-4 e^3 x^3 \log (x) \left (3 a+b n+3 b \log \left (c x^n\right )\right )+12 a \log (1+e x)+4 b n \log (1+e x)+12 a e^3 x^3 \log (1+e x)+4 b e^3 n x^3 \log (1+e x)+12 b \log \left (c x^n\right ) \log (1+e x)+12 b e^3 x^3 \log \left (c x^n\right ) \log (1+e x)+12 b e^3 n x^3 \text {Li}_2(-e x)}{36 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 796, normalized size = 4.08
method | result | size |
risch | \(-\frac {\ln \left (e x +1\right ) a}{3 x^{3}}+\left (-\frac {b \ln \left (e x +1\right )}{3 x^{3}}-\frac {b e \left (2 e^{2} \ln \left (e x +1\right ) x^{2}-2 e^{2} \ln \left (x \right ) x^{2}-2 e x +1\right )}{6 x^{2}}\right ) \ln \left (x^{n}\right )+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +1\right )}{6 x^{3}}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} e^{2}}{6 x}+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} e^{3} \ln \left (e x +1\right )}{6}+\frac {i e \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{12 x^{2}}-\frac {b \ln \left (c \right ) \ln \left (e x +1\right )}{3 x^{3}}+\frac {e^{3} b \ln \left (c \right ) \ln \left (e x \right )}{3}+\frac {n b \,e^{3} \ln \left (e x \right )}{9}-\frac {e a}{6 x^{2}}+\frac {e^{3} a \ln \left (e x \right )}{3}-\frac {e^{3} a \ln \left (e x +1\right )}{3}+\frac {e^{2} a}{3 x}-\frac {e^{3} b n \dilog \left (e x +1\right )}{3}+\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 ) e^{3} \ln \left (e x +1\right )}{6}-\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 ) e^{2}}{6 x}-\frac {i e^{3} \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 \right )}{6}+\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 )}{6 x^{3}}+\frac {i e \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{12 x^{2}}+\frac {b \ln \left (c \right ) e^{2}}{3 x}-\frac {e b \ln \left (c \right )}{6 x^{2}}-\frac {b \ln \left (c \right ) e^{3} \ln \left (e x +1\right )}{3}+\frac {4 b \,e^{2} n}{9 x}-\frac {5 b e n}{36 x^{2}}-\frac {b \,e^{3} n \ln \left (e x +1\right )}{9}-\frac {b n \ln \left (e x +1\right )}{9 x^{3}}-\frac {b \,e^{3} n \ln \left (x \right )^{2}}{6}+\frac {i e^{3} \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x \right )}{6}-\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 )}{6 x^{3}}+\frac {i e^{3} \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x \right )}{6}-\frac {i e \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12 x^{2}}-\frac {i e \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{12 x^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e^{3} \ln \left (e x +1\right )}{6}+\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e^{2}}{6 x}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e^{2}}{6 x}-\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} e^{3} \ln \left (e x +1\right )}{6}-\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 )}{6 x^{3}}-\frac {i e^{3} \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x \right )}{6}\) | \(796\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 208, normalized size = 1.07 \begin {gather*} -\frac {1}{3} \, {\left (\log \left (x e + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-x e\right )\right )} b n e^{3} - \frac {1}{9} \, {\left (b {\left (n + 3 \, \log \left (c\right )\right )} + 3 \, a\right )} e^{3} \log \left (x e + 1\right ) - \frac {6 \, b n x^{3} e^{3} \log \left (x\right )^{2} - 4 \, {\left (b {\left (n + 3 \, \log \left (c\right )\right )} + 3 \, a\right )} x^{3} e^{3} \log \left (x\right ) - 4 \, {\left (b {\left (4 \, n + 3 \, \log \left (c\right )\right )} + 3 \, a\right )} x^{2} e^{2} + {\left (b {\left (5 \, n + 6 \, \log \left (c\right )\right )} + 6 \, a\right )} x e - 4 \, {\left (3 \, b n x^{3} e^{3} \log \left (x\right ) - b {\left (n + 3 \, \log \left (c\right )\right )} - 3 \, a\right )} \log \left (x e + 1\right ) - 6 \, {\left (2 \, b x^{3} e^{3} \log \left (x\right ) + 2 \, b x^{2} e^{2} - b x e - 2 \, {\left (b x^{3} e^{3} + b\right )} \log \left (x e + 1\right )\right )} \log \left (x^{n}\right )}{36 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (e\,x+1\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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