Optimal. Leaf size=30 \[ \frac {x^2}{3 \left (x+\log \left (\frac {3}{\left (4-e^x\right )^2 x^3 \log (x)}\right )\right )} \]
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Rubi [F]
time = 1.68, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-4 x+e^x x+\left (-12 x-4 x^2+e^x \left (3 x+3 x^2\right )\right ) \log (x)+\left (-8 x+2 e^x x\right ) \log (x) \log \left (\frac {3}{\left (16 x^3-8 e^x x^3+e^{2 x} x^3\right ) \log (x)}\right )}{\left (-12 x^2+3 e^x x^2\right ) \log (x)+\left (-24 x+6 e^x x\right ) \log (x) \log \left (\frac {3}{\left (16 x^3-8 e^x x^3+e^{2 x} x^3\right ) \log (x)}\right )+\left (-12+3 e^x\right ) \log (x) \log ^2\left (\frac {3}{\left (16 x^3-8 e^x x^3+e^{2 x} x^3\right ) \log (x)}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (4-e^x-\log (x) \left (3 e^x (1+x)-4 (3+x)+2 \left (-4+e^x\right ) \log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )\right )}{3 \left (4-e^x\right ) \log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {x \left (4-e^x-\log (x) \left (3 e^x (1+x)-4 (3+x)+2 \left (-4+e^x\right ) \log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )\right )}{\left (4-e^x\right ) \log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {8 x^2}{\left (-4+e^x\right ) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2}+\frac {x+3 x \log (x)+3 x^2 \log (x)+2 x \log (x) \log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )}{\log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {x+3 x \log (x)+3 x^2 \log (x)+2 x \log (x) \log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )}{\log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx+\frac {8}{3} \int \frac {x^2}{\left (-4+e^x\right ) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {x+3 x \log (x)+x^2 \log (x)}{\log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2}+\frac {2 x}{x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )}\right ) \, dx+\frac {8}{3} \int \frac {x^2}{\left (-4+e^x\right ) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {x+3 x \log (x)+x^2 \log (x)}{\log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx+\frac {2}{3} \int \frac {x}{x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )} \, dx+\frac {8}{3} \int \frac {x^2}{\left (-4+e^x\right ) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {x+x (3+x) \log (x)}{\log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx+\frac {2}{3} \int \frac {x}{x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )} \, dx+\frac {8}{3} \int \frac {x^2}{\left (-4+e^x\right ) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {3 x}{\left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2}+\frac {x^2}{\left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2}+\frac {x}{\log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2}\right ) \, dx+\frac {2}{3} \int \frac {x}{x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )} \, dx+\frac {8}{3} \int \frac {x^2}{\left (-4+e^x\right ) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {x^2}{\left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx+\frac {1}{3} \int \frac {x}{\log (x) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx+\frac {2}{3} \int \frac {x}{x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )} \, dx+\frac {8}{3} \int \frac {x^2}{\left (-4+e^x\right ) \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx+\int \frac {x}{\left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 28, normalized size = 0.93 \begin {gather*} \frac {x^2}{3 \left (x+\log \left (\frac {3}{\left (-4+e^x\right )^2 x^3 \log (x)}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.67, size = 465, normalized size = 15.50
method | result | size |
risch | \(\frac {2 x^{2}}{3 \left (2 x +i \pi \,\mathrm {csgn}\left (\frac {i}{x^{3}}\right ) \mathrm {csgn}\left (\frac {i}{x^{3} \ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right )^{2}+i \pi \mathrm {csgn}\left (i x^{3}\right )^{3}-2 \ln \left (\ln \left (x \right )\right )+2 \ln \left (3\right )-6 \ln \left (x \right )+i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-4\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-4\right )^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i}{\ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right )^{3}-i \pi \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-4\right )^{2}\right )^{3}-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \ln \left ({\mathrm e}^{x}-4\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i}{x^{3} \ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right )^{3}-i \pi \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right )^{2}+i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-4\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-4\right )^{2}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{x^{3}}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{3} \ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{3} \ln \left (x \right ) \left ({\mathrm e}^{x}-4\right )^{2}}\right )^{2}\right )}\) | \(465\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 27, normalized size = 0.90 \begin {gather*} \frac {x^{2}}{3 \, {\left (x + \log \left (3\right ) - 3 \, \log \left (x\right ) - 2 \, \log \left (e^{x} - 4\right ) - \log \left (\log \left (x\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 39, normalized size = 1.30 \begin {gather*} \frac {x^{2}}{3 \, {\left (x + \log \left (\frac {3}{{\left (x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} + 16 \, x^{3}\right )} \log \left (x\right )}\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 36, normalized size = 1.20 \begin {gather*} \frac {x^{2}}{3 x + 3 \log {\left (\frac {3}{\left (x^{3} e^{2 x} - 8 x^{3} e^{x} + 16 x^{3}\right ) \log {\left (x \right )}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.52, size = 36, normalized size = 1.20 \begin {gather*} \frac {x^{2}}{3 \, {\left (x + \log \left (3\right ) - \log \left (e^{\left (2 \, x\right )} \log \left (x\right ) - 8 \, e^{x} \log \left (x\right ) + 16 \, \log \left (x\right )\right ) - 3 \, \log \left (x\right )\right )}} \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 = -0.03 \begin {gather*} \int -\frac {4\,x+\ln \left (x\right )\,\left (12\,x-{\mathrm {e}}^x\,\left (3\,x^2+3\,x\right )+4\,x^2\right )-x\,{\mathrm {e}}^x+\ln \left (\frac {3}{\ln \left (x\right )\,\left (x^3\,{\mathrm {e}}^{2\,x}-8\,x^3\,{\mathrm {e}}^x+16\,x^3\right )}\right )\,\ln \left (x\right )\,\left (8\,x-2\,x\,{\mathrm {e}}^x\right )}{\ln \left (x\right )\,\left (3\,{\mathrm {e}}^x-12\right )\,{\ln \left (\frac {3}{\ln \left (x\right )\,\left (x^3\,{\mathrm {e}}^{2\,x}-8\,x^3\,{\mathrm {e}}^x+16\,x^3\right )}\right )}^2-\ln \left (x\right )\,\left (24\,x-6\,x\,{\mathrm {e}}^x\right )\,\ln \left (\frac {3}{\ln \left (x\right )\,\left (x^3\,{\mathrm {e}}^{2\,x}-8\,x^3\,{\mathrm {e}}^x+16\,x^3\right )}\right )+\ln \left (x\right )\,\left (3\,x^2\,{\mathrm {e}}^x-12\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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