Optimal. Leaf size=24 \[ \left (-e^2+\log (x)+x (2-3 \log (4)-4 \log (16)+\log (x))\right )^2 \]
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Rubi [B] time = 0.20, antiderivative size = 160, normalized size of antiderivative = 6.67, number of steps used = 17, number of rules used = 10, integrand size = 140, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6, 14, 76, 2357, 2301, 2304, 2295, 2330, 2296, 2305} \begin {gather*} \frac {x^2}{2}+x^2 \log ^2(x)+x^2 (5-6 \log (4)-8 \log (16)) \log (x)-x^2 \log (x)+x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-\frac {1}{2} x^2 (5-6 \log (4)-8 \log (16))+4 x+2 x \log ^2(x)+\log ^2(x)+2 x \left (4-e^2-\log (4194304)\right ) \log (x)-4 x \log (x)+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 x \left (4-e^2-\log (4194304)\right )-2 e^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 14
Rule 76
Rule 2295
Rule 2296
Rule 2301
Rule 2304
Rule 2305
Rule 2330
Rule 2357
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^2 (-2-6 x)+4 x+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+x^2 \left (12+18 \log ^2(4)\right )+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx\\ &=\int \frac {e^2 (-2-6 x)+4 x+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+x^2 \left (12+18 \log ^2(4)+32 \log ^2(16)\right )+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx\\ &=\int \left (\frac {2 \left (e^2-x (2-4 \log (16)-\log (64))\right ) (-1-x (3-4 \log (16)-\log (64)))}{x}+\frac {2 \left (1+x^2 (5-6 \log (4)-8 \log (16))+x \left (4-e^2-\log (4194304)\right )\right ) \log (x)}{x}+2 (1+x) \log ^2(x)\right ) \, dx\\ &=2 \int \frac {\left (e^2-x (2-4 \log (16)-\log (64))\right ) (-1-x (3-4 \log (16)-\log (64)))}{x} \, dx+2 \int \frac {\left (1+x^2 (5-6 \log (4)-8 \log (16))+x \left (4-e^2-\log (4194304)\right )\right ) \log (x)}{x} \, dx+2 \int (1+x) \log ^2(x) \, dx\\ &=2 \int \left (2-\frac {e^2}{x}-e^2 (3-4 \log (16)-\log (64))+x (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-\log (4194304)\right ) \, dx+2 \int \left (\frac {\log (x)}{x}+x (5-6 \log (4)-8 \log (16)) \log (x)+\left (4-e^2-\log (4194304)\right ) \log (x)\right ) \, dx+2 \int \left (\log ^2(x)+x \log ^2(x)\right ) \, dx\\ &=x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 e^2 \log (x)+2 \int \frac {\log (x)}{x} \, dx+2 \int \log ^2(x) \, dx+2 \int x \log ^2(x) \, dx+(2 (5-6 \log (4)-8 \log (16))) \int x \log (x) \, dx+\left (2 \left (4-e^2-\log (4194304)\right )\right ) \int \log (x) \, dx\\ &=-\frac {1}{2} x^2 (5-6 \log (4)-8 \log (16))+x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-2 x \left (4-e^2-\log (4194304)\right )+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 e^2 \log (x)+x^2 (5-6 \log (4)-8 \log (16)) \log (x)+2 x \left (4-e^2-\log (4194304)\right ) \log (x)+\log ^2(x)+2 x \log ^2(x)+x^2 \log ^2(x)-2 \int x \log (x) \, dx-4 \int \log (x) \, dx\\ &=4 x+\frac {x^2}{2}-\frac {1}{2} x^2 (5-6 \log (4)-8 \log (16))+x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-2 x \left (4-e^2-\log (4194304)\right )+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 e^2 \log (x)-4 x \log (x)-x^2 \log (x)+x^2 (5-6 \log (4)-8 \log (16)) \log (x)+2 x \left (4-e^2-\log (4194304)\right ) \log (x)+\log ^2(x)+2 x \log ^2(x)+x^2 \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.16, size = 152, normalized size = 6.33 \begin {gather*} 2 \left (-2 e^2 x+2 x^2+\frac {1}{4} x^2 \log (16)+\frac {5}{4} x^2 \log (256)+2 e^2 x \log (2048)-\frac {5}{2} x^2 \log (4194304)+\frac {1}{2} x^2 \log (4) \log (4194304)+\frac {5}{2} x^2 \log (16) \log (4194304)-e^2 \log (x)+2 x \log (x)-e^2 x \log (x)+2 x^2 \log (x)-\frac {1}{2} x^2 \log (16) \log (x)-\frac {5}{2} x^2 \log (256) \log (x)-2 x \log (2048) \log (x)+\frac {\log ^2(x)}{2}+x \log ^2(x)+\frac {1}{2} x^2 \log ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.47, size = 77, normalized size = 3.21 \begin {gather*} 484 \, x^{2} \log \relax (2)^{2} + {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} + 4 \, x^{2} - 4 \, x e^{2} - 44 \, {\left (2 \, x^{2} - x e^{2}\right )} \log \relax (2) + 2 \, {\left (2 \, x^{2} - {\left (x + 1\right )} e^{2} - 22 \, {\left (x^{2} + x\right )} \log \relax (2) + 2 \, x\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 94, normalized size = 3.92 \begin {gather*} 484 \, x^{2} \log \relax (2)^{2} - 44 \, x^{2} \log \relax (2) \log \relax (x) + x^{2} \log \relax (x)^{2} - 88 \, x^{2} \log \relax (2) + 44 \, x e^{2} \log \relax (2) + 4 \, x^{2} \log \relax (x) - 2 \, x e^{2} \log \relax (x) - 44 \, x \log \relax (2) \log \relax (x) + 2 \, x \log \relax (x)^{2} + 4 \, x^{2} - 4 \, x e^{2} + 4 \, x \log \relax (x) - 2 \, e^{2} \log \relax (x) + \log \relax (x)^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 82, normalized size = 3.42
method | result | size |
norman | \(\ln \relax (x )^{2}+x^{2} \ln \relax (x )^{2}-2 \,{\mathrm e}^{2} \ln \relax (x )+\left (-4 \,{\mathrm e}^{2}+44 \,{\mathrm e}^{2} \ln \relax (2)\right ) x +\left (4-88 \ln \relax (2)+484 \ln \relax (2)^{2}\right ) x^{2}+\left (4-44 \ln \relax (2)\right ) x^{2} \ln \relax (x )+\left (4-2 \,{\mathrm e}^{2}-44 \ln \relax (2)\right ) x \ln \relax (x )+2 x \ln \relax (x )^{2}\) | \(82\) |
risch | \(\left (x^{2}+2 x +1\right ) \ln \relax (x )^{2}+\left (-44 x^{2} \ln \relax (2)-2 \,{\mathrm e}^{2} x -44 x \ln \relax (2)+4 x^{2}+4 x \right ) \ln \relax (x )+484 x^{2} \ln \relax (2)^{2}+44 x \,{\mathrm e}^{2} \ln \relax (2)-88 x^{2} \ln \relax (2)-4 \,{\mathrm e}^{2} x +4 x^{2}-2 \,{\mathrm e}^{2} \ln \relax (x )\) | \(83\) |
default | \(x^{2} \ln \relax (x )^{2}+4 x^{2} \ln \relax (x )+4 x^{2}-88 \ln \relax (2) \left (\frac {x^{2} \ln \relax (x )}{2}-\frac {x^{2}}{4}\right )+484 x^{2} \ln \relax (2)^{2}-2 \,{\mathrm e}^{2} \left (x \ln \relax (x )-x \right )+44 x \,{\mathrm e}^{2} \ln \relax (2)+2 x \ln \relax (x )^{2}+4 x \ln \relax (x )-44 \ln \relax (2) \left (x \ln \relax (x )-x \right )-110 x^{2} \ln \relax (2)-6 \,{\mathrm e}^{2} x -44 x \ln \relax (2)-2 \,{\mathrm e}^{2} \ln \relax (x )+\ln \relax (x )^{2}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 135, normalized size = 5.62 \begin {gather*} 484 \, x^{2} \log \relax (2)^{2} + \frac {1}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} - 110 \, x^{2} \log \relax (2) + 44 \, x e^{2} \log \relax (2) + 5 \, x^{2} \log \relax (x) + 2 \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + \frac {7}{2} \, x^{2} - 2 \, {\left (x \log \relax (x) - x\right )} e^{2} - 6 \, x e^{2} - 22 \, {\left (2 \, x^{2} \log \relax (x) - x^{2}\right )} \log \relax (2) - 44 \, {\left (x \log \relax (x) - x\right )} \log \relax (2) - 44 \, x \log \relax (2) + 8 \, x \log \relax (x) - 2 \, e^{2} \log \relax (x) + \log \relax (x)^{2} - 4 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 35, normalized size = 1.46 \begin {gather*} \left (2\,x+\ln \relax (x)-22\,x\,\ln \relax (2)+x\,\ln \relax (x)\right )\,\left (2\,x-2\,{\mathrm {e}}^2+\ln \relax (x)-22\,x\,\ln \relax (2)+x\,\ln \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.35, size = 87, normalized size = 3.62 \begin {gather*} x^{2} \left (- 88 \log {\relax (2 )} + 4 + 484 \log {\relax (2 )}^{2}\right ) + x \left (- 4 e^{2} + 44 e^{2} \log {\relax (2 )}\right ) + \left (x^{2} + 2 x + 1\right ) \log {\relax (x )}^{2} + \left (- 44 x^{2} \log {\relax (2 )} + 4 x^{2} - 44 x \log {\relax (2 )} - 2 x e^{2} + 4 x\right ) \log {\relax (x )} - 2 e^{2} \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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