Optimal. Leaf size=33 \[ \frac {1}{9} \left (3+e^{e^x}-x\right )^2 x^2 \left (\frac {3}{4}+\log \left (\frac {4}{x}\right )\right ) \log ^2(x) \]
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Rubi [F] time = 40.85, 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 {1}{18} \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{18} \int \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx\\ &=\frac {1}{18} \int \left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x) \, dx+\frac {1}{18} \int \left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \, dx+\frac {1}{18} \int e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right ) \, dx+\frac {1}{18} \int e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right ) \, dx\\ &=\frac {1}{18} \int (3-x)^2 x \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x) \, dx+\frac {1}{18} \int (3-x) x \left (3-4 x-4 (-3+2 x) \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \, dx+\frac {1}{18} \int e^{e^x} x \log (x) \left (18-6 x-8 (-3+x) \log \left (\frac {4}{x}\right )+\left (6-5 x-3 e^x (-3+x) x-4 \left (-6-3 \left (-1+e^x\right ) x+e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)\right ) \, dx+\frac {1}{18} \int e^{2 e^x} x \log (x) \left (3+\log (x)+3 e^x x \log (x)+4 \log \left (\frac {4}{x}\right ) \left (1+\log (x)+e^x x \log (x)\right )\right ) \, dx\\ &=\frac {1}{18} \int \left (9 x \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)-6 x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)+x^3 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)\right ) \, dx+\frac {1}{18} \int \left (-3 x \left (-3+4 x-12 \log \left (\frac {4}{x}\right )+8 x \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+x^2 \left (-3+4 x-12 \log \left (\frac {4}{x}\right )+8 x \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right ) \, dx+\frac {1}{18} \int \left (e^{2 e^x+x} x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} x \log (x) \left (3+4 \log \left (\frac {4}{x}\right )+\log (x)+4 \log \left (\frac {4}{x}\right ) \log (x)\right )\right ) \, dx+\frac {1}{18} \int \left (-e^{e^x+x} (-3+x) x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x)-e^{e^x} x \log (x) \left (-18+6 x-24 \log \left (\frac {4}{x}\right )+8 x \log \left (\frac {4}{x}\right )-6 \log (x)+5 x \log (x)-24 \log \left (\frac {4}{x}\right ) \log (x)+12 x \log \left (\frac {4}{x}\right ) \log (x)\right )\right ) \, dx\\ &=\frac {1}{18} \int x^3 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x) \, dx+\frac {1}{18} \int e^{2 e^x+x} x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \, dx-\frac {1}{18} \int e^{e^x+x} (-3+x) x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \, dx+\frac {1}{18} \int x^2 \left (-3+4 x-12 \log \left (\frac {4}{x}\right )+8 x \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \, dx+\frac {1}{18} \int e^{2 e^x} x \log (x) \left (3+4 \log \left (\frac {4}{x}\right )+\log (x)+4 \log \left (\frac {4}{x}\right ) \log (x)\right ) \, dx-\frac {1}{18} \int e^{e^x} x \log (x) \left (-18+6 x-24 \log \left (\frac {4}{x}\right )+8 x \log \left (\frac {4}{x}\right )-6 \log (x)+5 x \log (x)-24 \log \left (\frac {4}{x}\right ) \log (x)+12 x \log \left (\frac {4}{x}\right ) \log (x)\right ) \, dx-\frac {1}{6} \int x \left (-3+4 x-12 \log \left (\frac {4}{x}\right )+8 x \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \, dx-\frac {1}{3} \int x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x) \, dx+\frac {1}{2} \int x \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x) \, dx\\ &=\frac {1}{2} x^2 \log (x)-\frac {4}{27} x^3 \log (x)+\frac {1}{72} x^4 \log (x)+\frac {1}{4} x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)-\frac {1}{9} x^3 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)+\frac {1}{72} x^4 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)-\frac {1}{18} \int x^3 \left (1+\log \left (\frac {4}{x}\right )\right ) \, dx+\frac {1}{18} \int \left (3 e^{2 e^x+x} x^2 \log ^2(x)+4 e^{2 e^x+x} x^2 \log \left (\frac {4}{x}\right ) \log ^2(x)\right ) \, dx+\frac {1}{18} \int \left (-3 x^2 \log ^2(x)+4 x^3 \log ^2(x)-12 x^2 \log \left (\frac {4}{x}\right ) \log ^2(x)+8 x^3 \log \left (\frac {4}{x}\right ) \log ^2(x)\right ) \, dx-\frac {1}{18} \int \left (-3 e^{e^x+x} x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{e^x+x} x^3 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right ) \, dx+\frac {1}{18} \int e^{2 e^x} x \log (x) \left (3+\log (x)+4 \log \left (\frac {4}{x}\right ) (1+\log (x))\right ) \, dx-\frac {1}{18} \int e^{e^x} x \log (x) \left (6 (-3+x)+(-6+5 x) \log (x)+4 \log \left (\frac {4}{x}\right ) (2 (-3+x)+3 (-2+x) \log (x))\right ) \, dx-\frac {1}{6} \int \left (-3 x \log ^2(x)+4 x^2 \log ^2(x)-12 x \log \left (\frac {4}{x}\right ) \log ^2(x)+8 x^2 \log \left (\frac {4}{x}\right ) \log ^2(x)\right ) \, dx+\frac {1}{3} \int \frac {1}{9} x^2 \left (13+12 \log \left (\frac {4}{x}\right )\right ) \, dx-\frac {1}{2} \int \frac {1}{2} x \left (5+4 \log \left (\frac {4}{x}\right )\right ) \, dx\\ &=-\frac {x^4}{288}-\frac {1}{72} x^4 \left (1+\log \left (\frac {4}{x}\right )\right )+\frac {1}{2} x^2 \log (x)-\frac {4}{27} x^3 \log (x)+\frac {1}{72} x^4 \log (x)+\frac {1}{4} x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)-\frac {1}{9} x^3 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)+\frac {1}{72} x^4 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)+\frac {1}{27} \int x^2 \left (13+12 \log \left (\frac {4}{x}\right )\right ) \, dx-\frac {1}{18} \int e^{e^x+x} x^3 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \, dx+\frac {1}{18} \int \left (e^{2 e^x} x \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)+e^{2 e^x} x \left (1+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right ) \, dx-\frac {1}{18} \int \left (2 e^{e^x} (-3+x) x \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log (x)+e^{e^x} x \left (-6+5 x-24 \log \left (\frac {4}{x}\right )+12 x \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right ) \, dx-\frac {1}{6} \int x^2 \log ^2(x) \, dx+\frac {1}{6} \int e^{2 e^x+x} x^2 \log ^2(x) \, dx+\frac {1}{6} \int e^{e^x+x} x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \, dx+\frac {2}{9} \int x^3 \log ^2(x) \, dx+\frac {2}{9} \int e^{2 e^x+x} x^2 \log \left (\frac {4}{x}\right ) \log ^2(x) \, dx-\frac {1}{4} \int x \left (5+4 \log \left (\frac {4}{x}\right )\right ) \, dx+\frac {4}{9} \int x^3 \log \left (\frac {4}{x}\right ) \log ^2(x) \, dx+\frac {1}{2} \int x \log ^2(x) \, dx-\frac {2}{3} \int x^2 \log ^2(x) \, dx-\frac {2}{3} \int x^2 \log \left (\frac {4}{x}\right ) \log ^2(x) \, dx-\frac {4}{3} \int x^2 \log \left (\frac {4}{x}\right ) \log ^2(x) \, dx+2 \int x \log \left (\frac {4}{x}\right ) \log ^2(x) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.20, size = 33, normalized size = 1.00 \begin {gather*} \frac {1}{36} \left (3+e^{e^x}-x\right )^2 x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 324, normalized size = 9.82 \begin {gather*} \frac {1}{9} \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (\frac {4}{x}\right )^{3} + \frac {1}{3} \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \relax (2)^{2} + \frac {1}{36} \, {\left (3 \, x^{4} - 18 \, x^{3} + 27 \, x^{2} - 16 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \relax (2)\right )} \log \left (\frac {4}{x}\right )^{2} + \frac {1}{36} \, {\left (4 \, x^{2} \log \left (\frac {4}{x}\right )^{3} + 12 \, x^{2} \log \relax (2)^{2} - {\left (16 \, x^{2} \log \relax (2) - 3 \, x^{2}\right )} \log \left (\frac {4}{x}\right )^{2} + 4 \, {\left (4 \, x^{2} \log \relax (2)^{2} - 3 \, x^{2} \log \relax (2)\right )} \log \left (\frac {4}{x}\right )\right )} e^{\left (2 \, e^{x}\right )} - \frac {1}{18} \, {\left (4 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (\frac {4}{x}\right )^{3} + 12 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \relax (2)^{2} + {\left (3 \, x^{3} - 9 \, x^{2} - 16 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \relax (2)\right )} \log \left (\frac {4}{x}\right )^{2} + 4 \, {\left (4 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \relax (2)^{2} - 3 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \relax (2)\right )} \log \left (\frac {4}{x}\right )\right )} e^{\left (e^{x}\right )} + \frac {1}{9} \, {\left (4 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \relax (2)^{2} - 3 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \relax (2)\right )} \log \left (\frac {4}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.54, size = 227, normalized size = 6.88 \begin {gather*} \frac {2}{9} \, x^{4} \log \relax (2) \log \relax (x)^{2} - \frac {1}{9} \, x^{4} \log \relax (x)^{3} + \frac {1}{12} \, x^{4} \log \relax (x)^{2} - \frac {4}{3} \, x^{3} \log \relax (2) \log \relax (x)^{2} + \frac {2}{9} \, x^{2} e^{\left (2 \, e^{x}\right )} \log \relax (2) \log \relax (x)^{2} + \frac {2}{3} \, x^{3} \log \relax (x)^{3} - \frac {1}{9} \, x^{2} e^{\left (2 \, e^{x}\right )} \log \relax (x)^{3} - \frac {1}{2} \, x^{3} \log \relax (x)^{2} + \frac {1}{12} \, x^{2} e^{\left (2 \, e^{x}\right )} \log \relax (x)^{2} + 2 \, x^{2} \log \relax (2) \log \relax (x)^{2} - x^{2} \log \relax (x)^{3} + \frac {3}{4} \, x^{2} \log \relax (x)^{2} - \frac {1}{18} \, {\left (8 \, x^{3} e^{\left (x + e^{x}\right )} \log \relax (2) \log \relax (x)^{2} - 4 \, x^{3} e^{\left (x + e^{x}\right )} \log \relax (x)^{3} + 3 \, x^{3} e^{\left (x + e^{x}\right )} \log \relax (x)^{2} - 24 \, x^{2} e^{\left (x + e^{x}\right )} \log \relax (2) \log \relax (x)^{2} + 12 \, x^{2} e^{\left (x + e^{x}\right )} \log \relax (x)^{3} - 9 \, x^{2} e^{\left (x + e^{x}\right )} \log \relax (x)^{2}\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.68, size = 209, normalized size = 6.33
method | result | size |
risch | \(-\frac {x^{3} \ln \relax (x )^{2}}{2}+\frac {4 \ln \relax (x )^{2} {\mathrm e}^{{\mathrm e}^{x}} \ln \relax (2) x^{2}}{3}+\frac {3 x^{2} \ln \relax (x )^{2}}{4}+\frac {\ln \relax (x )^{2} {\mathrm e}^{{\mathrm e}^{x}} x^{2}}{2}-\frac {4 \ln \relax (x )^{2} \ln \relax (2) x^{3}}{3}+2 x^{2} \ln \relax (2) \ln \relax (x )^{2}+\frac {x^{4} \ln \relax (x )^{2}}{12}+\frac {2 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} \ln \relax (x )^{2} \ln \relax (2) x^{2}}{9}-\frac {4 \ln \relax (x )^{2} {\mathrm e}^{{\mathrm e}^{x}} \ln \relax (2) x^{3}}{9}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}} \ln \relax (x )^{2} x^{2}}{12}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}} x^{2} \ln \relax (x )^{3}}{9}+\frac {2 \ln \relax (x )^{2} \ln \relax (2) x^{4}}{9}+\frac {2 \ln \relax (x )^{3} {\mathrm e}^{{\mathrm e}^{x}} x^{3}}{9}-\frac {2 \ln \relax (x )^{3} {\mathrm e}^{{\mathrm e}^{x}} x^{2}}{3}-\frac {\ln \relax (x )^{2} {\mathrm e}^{{\mathrm e}^{x}} x^{3}}{6}-x^{2} \ln \relax (x )^{3}+\frac {2 x^{3} \ln \relax (x )^{3}}{3}-\frac {x^{4} \ln \relax (x )^{3}}{9}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 223, normalized size = 6.76 \begin {gather*} \frac {1}{36} \, {\left (3 \, x^{4} - 18 \, x^{3} + 27 \, x^{2} + 4 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (\frac {4}{x}\right )\right )} \log \relax (x)^{2} + \frac {1}{36} \, {\left (x^{2} {\left (8 \, \log \relax (2) + 3\right )} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (x)^{3}\right )} e^{\left (2 \, e^{x}\right )} + \frac {1}{18} \, {\left (4 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \relax (x)^{3} - {\left (x^{3} {\left (8 \, \log \relax (2) + 3\right )} - 3 \, x^{2} {\left (8 \, \log \relax (2) + 3\right )}\right )} \log \relax (x)^{2}\right )} e^{\left (e^{x}\right )} - \frac {1}{108} \, {\left (6 \, x^{4} {\left (2 \, \log \relax (2) + 1\right )} - 4 \, x^{3} {\left (24 \, \log \relax (2) + 13\right )} + 27 \, x^{2} {\left (8 \, \log \relax (2) + 5\right )} - 6 \, {\left (x^{4} - 8 \, x^{3} + 18 \, x^{2}\right )} \log \relax (x)\right )} \log \relax (x) + \frac {1}{108} \, {\left (6 \, x^{4} - 52 \, x^{3} + 135 \, x^{2} + 6 \, {\left (x^{4} - 8 \, x^{3} + 18 \, x^{2}\right )} \log \left (\frac {4}{x}\right )\right )} \log \relax (x) \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 {{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (\left (x+3\,x^2\,{\mathrm {e}}^x+\ln \left (\frac {4}{x}\right )\,\left (4\,x+4\,x^2\,{\mathrm {e}}^x\right )\right )\,{\ln \relax (x)}^2+\left (3\,x+4\,x\,\ln \left (\frac {4}{x}\right )\right )\,\ln \relax (x)\right )}{18}+\frac {\ln \relax (x)\,\left (27\,x+\ln \left (\frac {4}{x}\right )\,\left (4\,x^3-24\,x^2+36\,x\right )-18\,x^2+3\,x^3\right )}{18}+\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (\left (6\,x+{\mathrm {e}}^x\,\left (9\,x^2-3\,x^3\right )+\ln \left (\frac {4}{x}\right )\,\left (24\,x+{\mathrm {e}}^x\,\left (12\,x^2-4\,x^3\right )-12\,x^2\right )-5\,x^2\right )\,{\ln \relax (x)}^2+\left (18\,x+\ln \left (\frac {4}{x}\right )\,\left (24\,x-8\,x^2\right )-6\,x^2\right )\,\ln \relax (x)\right )}{18}+\frac {{\ln \relax (x)}^2\,\left (9\,x+\ln \left (\frac {4}{x}\right )\,\left (8\,x^3-36\,x^2+36\,x\right )-15\,x^2+4\,x^3\right )}{18} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.33, size = 187, normalized size = 5.67 \begin {gather*} \left (- \frac {x^{4}}{9} + \frac {2 x^{3}}{3} - x^{2}\right ) \log {\relax (x )}^{3} + \frac {\left (- 72 x^{2} \log {\relax (x )}^{3} + 54 x^{2} \log {\relax (x )}^{2} + 144 x^{2} \log {\relax (2 )} \log {\relax (x )}^{2}\right ) e^{2 e^{x}}}{648} + \left (\frac {x^{4}}{12} + \frac {2 x^{4} \log {\relax (2 )}}{9} - \frac {4 x^{3} \log {\relax (2 )}}{3} - \frac {x^{3}}{2} + \frac {3 x^{2}}{4} + 2 x^{2} \log {\relax (2 )}\right ) \log {\relax (x )}^{2} + \frac {\left (144 x^{3} \log {\relax (x )}^{3} - 288 x^{3} \log {\relax (2 )} \log {\relax (x )}^{2} - 108 x^{3} \log {\relax (x )}^{2} - 432 x^{2} \log {\relax (x )}^{3} + 324 x^{2} \log {\relax (x )}^{2} + 864 x^{2} \log {\relax (2 )} \log {\relax (x )}^{2}\right ) e^{e^{x}}}{648} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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