Optimal. Leaf size=31 \[ 2 \log \left (\frac {\frac {16 e^{8 x}}{x^2}+x}{\log \left (3-\log \left (\frac {3}{20+x}\right )\right )}\right ) \]
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Rubi [F] time = 7.84, 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 {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\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 {2 \left (-16 e^{8 x} x-x^4-(20+x) \left (x^3+32 e^{8 x} (-1+4 x)\right ) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )}{x (20+x) \left (16 e^{8 x}+x^3\right ) \left (3-\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx\\ &=2 \int \frac {-16 e^{8 x} x-x^4-(20+x) \left (x^3+32 e^{8 x} (-1+4 x)\right ) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{x (20+x) \left (16 e^{8 x}+x^3\right ) \left (3-\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx\\ &=2 \int \left (-\frac {x^2 (-3+8 x)}{16 e^{8 x}+x^3}+\frac {x+120 \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-474 x \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-24 x^2 \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-40 \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )+158 x \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )+8 x^2 \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{x (20+x) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x^2 (-3+8 x)}{16 e^{8 x}+x^3} \, dx\right )+2 \int \frac {x+120 \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-474 x \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-24 x^2 \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-40 \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )+158 x \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )+8 x^2 \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{x (20+x) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx\\ &=-\left (2 \int \left (-\frac {3 x^2}{16 e^{8 x}+x^3}+\frac {8 x^3}{16 e^{8 x}+x^3}\right ) \, dx\right )+2 \int \frac {158-\frac {40}{x}+8 x+\frac {1}{\left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}}{20+x} \, dx\\ &=2 \int \left (\frac {2 (-1+4 x)}{x}+\frac {1}{(20+x) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}\right ) \, dx+6 \int \frac {x^2}{16 e^{8 x}+x^3} \, dx-16 \int \frac {x^3}{16 e^{8 x}+x^3} \, dx\\ &=2 \int \frac {1}{(20+x) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx+4 \int \frac {-1+4 x}{x} \, dx+6 \int \frac {x^2}{16 e^{8 x}+x^3} \, dx-16 \int \frac {x^3}{16 e^{8 x}+x^3} \, dx\\ &=-2 \log \left (\log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )+4 \int \left (4-\frac {1}{x}\right ) \, dx+6 \int \frac {x^2}{16 e^{8 x}+x^3} \, dx-16 \int \frac {x^3}{16 e^{8 x}+x^3} \, dx\\ &=16 x-4 \log (x)-2 \log \left (\log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )+6 \int \frac {x^2}{16 e^{8 x}+x^3} \, dx-16 \int \frac {x^3}{16 e^{8 x}+x^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 35, normalized size = 1.13 \begin {gather*} 2 \left (-2 \log (x)+\log \left (16 e^{8 x}+x^3\right )-\log \left (\log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 34, normalized size = 1.10 \begin {gather*} 2 \, \log \left (x^{3} + 16 \, e^{\left (8 \, x\right )}\right ) - 4 \, \log \relax (x) - 2 \, \log \left (\log \left (-\log \left (\frac {3}{x + 20}\right ) + 3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 34, normalized size = 1.10 \begin {gather*} 2 \, \log \left (-x^{3} - 16 \, e^{\left (8 \, x\right )}\right ) - 4 \, \log \relax (x) - 2 \, \log \left (\log \left (-\log \relax (3) + \log \left (x + 20\right ) + 3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 33, normalized size = 1.06
method | result | size |
risch | \(-4 \ln \relax (x )+2 \ln \left (\frac {x^{3}}{16}+{\mathrm e}^{8 x}\right )-2 \ln \left (\ln \left (-\ln \relax (3)+\ln \left (20+x \right )+3\right )\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 32, normalized size = 1.03 \begin {gather*} 2 \, \log \left (\frac {1}{16} \, x^{3} + e^{\left (8 \, x\right )}\right ) - 4 \, \log \relax (x) - 2 \, \log \left (\log \left (-\log \relax (3) + \log \left (x + 20\right ) + 3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 34, normalized size = 1.10 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^{8\,x}+\frac {x^3}{16}\right )-2\,\ln \left (\ln \left (3-\ln \left (\frac {3}{x+20}\right )\right )\right )-4\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 31, normalized size = 1.00 \begin {gather*} - 4 \log {\relax (x )} + 2 \log {\left (\frac {x^{3}}{16} + e^{8 x} \right )} - 2 \log {\left (\log {\left (3 - \log {\left (\frac {3}{x + 20} \right )} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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