Optimal. Leaf size=28 \[ 2 x+\left (\log (5)-4 \left (x^2+\log \left (e^{e^3}+\log (2+x)\right )\right )^2\right )^2 \]
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Rubi [F] time = 36.89, 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 {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx\\ &=\int \left (\frac {2 \left (2 e^{e^3}+e^{e^3} x+32 x^6+128 e^{e^3} x^7+64 e^{e^3} x^8-8 x^2 \log (5)-32 e^{e^3} x^3 \log (5)-16 e^{e^3} x^4 \log (5)+2 \log (2+x)+x \log (2+x)+128 x^7 \log (2+x)+64 x^8 \log (2+x)-32 x^3 \log (5) \log (2+x)-16 x^4 \log (5) \log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}+\frac {16 \left (12 x^4-\log (5)\right ) \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}+\frac {192 x^2 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}+\frac {64 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}\right ) \, dx\\ &=2 \int \frac {2 e^{e^3}+e^{e^3} x+32 x^6+128 e^{e^3} x^7+64 e^{e^3} x^8-8 x^2 \log (5)-32 e^{e^3} x^3 \log (5)-16 e^{e^3} x^4 \log (5)+2 \log (2+x)+x \log (2+x)+128 x^7 \log (2+x)+64 x^8 \log (2+x)-32 x^3 \log (5) \log (2+x)-16 x^4 \log (5) \log (2+x)}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+16 \int \frac {\left (12 x^4-\log (5)\right ) \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+64 \int \frac {\left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+192 \int \frac {x^2 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx\\ &=2 \int \frac {32 x^6-8 x^2 \log (5)+e^{e^3} (2+x) \left (1+64 x^7-16 x^3 \log (5)\right )+(2+x) \left (1+64 x^7-16 x^3 \log (5)\right ) \log (2+x)}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+16 \int \frac {\left (12 x^4-\log (5)\right ) \left (1+2 e^{e^3} x (2+x)+2 x (2+x) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+64 \int \frac {\left (1+2 e^{e^3} x (2+x)+2 x (2+x) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+192 \int \frac {x^2 \left (1+2 e^{e^3} x (2+x)+2 x (2+x) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx\\ &=2 \int \left (1+64 x^7-16 x^3 \log (5)+\frac {8 x^2 \left (4 x^4-\log (5)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}\right ) \, dx+16 \int \left (-\frac {96 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)}+\frac {48 x \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)}-\frac {24 x^2 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)}+\frac {12 x^3 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)}-\frac {(-192+\log (5)) \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}\right ) \, dx+64 \int \left (\frac {\log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}+\frac {4 e^{e^3} x \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}+\frac {2 e^{e^3} x^2 \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}+\frac {4 x \log (2+x) \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}+\frac {2 x^2 \log (2+x) \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}\right ) \, dx+192 \int \left (-\frac {2 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)}+\frac {x \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)}+\frac {4 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )}\right ) \, dx\\ &=2 x+16 x^8-8 x^4 \log (5)+16 \int \frac {x^2 \left (4 x^4-\log (5)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+64 \int \frac {\log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+128 \int \frac {x^2 \log (2+x) \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+192 \int \frac {x^3 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)} \, dx+192 \int \frac {x \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)} \, dx+256 \int \frac {x \log (2+x) \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx-384 \int \frac {x^2 \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)} \, dx-384 \int \frac {\left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)} \, dx+768 \int \frac {x \left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)} \, dx+768 \int \frac {\left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx-1536 \int \frac {\left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{e^{e^3}+\log (2+x)} \, dx+\left (128 e^{e^3}\right ) \int \frac {x^2 \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+\left (256 e^{e^3}\right ) \int \frac {x \log ^3\left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx+(16 (192-\log (5))) \int \frac {\left (1+4 e^{e^3} x+2 e^{e^3} x^2+4 x \log (2+x)+2 x^2 \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )}{(2+x) \left (e^{e^3}+\log (2+x)\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.25, size = 100, normalized size = 3.57 \begin {gather*} 2 \left (x+8 x^8-4 x^4 \log (5)+8 x^2 \left (4 x^4-\log (5)\right ) \log \left (e^{e^3}+\log (2+x)\right )+4 \left (12 x^4-\log (5)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+32 x^2 \log ^3\left (e^{e^3}+\log (2+x)\right )+8 \log ^4\left (e^{e^3}+\log (2+x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.04, size = 92, normalized size = 3.29 \begin {gather*} 16 \, x^{8} - 8 \, x^{4} \log \relax (5) + 64 \, x^{2} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{3} + 16 \, \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{4} + 8 \, {\left (12 \, x^{4} - \log \relax (5)\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{2} + 16 \, {\left (4 \, x^{6} - x^{2} \log \relax (5)\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 106, normalized size = 3.79 \begin {gather*} 16 \, x^{8} + 64 \, x^{6} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 96 \, x^{4} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{2} - 8 \, x^{4} \log \relax (5) + 64 \, x^{2} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{3} - 16 \, x^{2} \log \relax (5) \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 16 \, \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{4} - 8 \, \log \relax (5) \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{2} + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 91, normalized size = 3.25
method | result | size |
risch | \(16 \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{4}+64 x^{2} \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{3}+\left (96 x^{4}-8 \ln \relax (5)\right ) \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{2}+\left (64 x^{6}-16 x^{2} \ln \relax (5)\right ) \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )+16 x^{8}-8 x^{4} \ln \relax (5)+2 x\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 111, normalized size = 3.96 \begin {gather*} 16 \, x^{8} - 8 \, x^{4} \log \relax (5) + 64 \, x^{2} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{3} + 16 \, \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{4} + 8 \, {\left (12 \, x^{4} - \log \relax (5)\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{2} + 4 \, {\left (16 \, x^{6} - 4 \, x^{2} \log \relax (5) - e^{\left (e^{3}\right )}\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 4 \, e^{\left (e^{3}\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.36, size = 92, normalized size = 3.29 \begin {gather*} 2\,x+16\,{\ln \left (\ln \left (x+2\right )+{\mathrm {e}}^{{\mathrm {e}}^3}\right )}^4+64\,x^2\,{\ln \left (\ln \left (x+2\right )+{\mathrm {e}}^{{\mathrm {e}}^3}\right )}^3-\ln \left (\ln \left (x+2\right )+{\mathrm {e}}^{{\mathrm {e}}^3}\right )\,\left (16\,x^2\,\ln \relax (5)-64\,x^6\right )-8\,x^4\,\ln \relax (5)+16\,x^8-{\ln \left (\ln \left (x+2\right )+{\mathrm {e}}^{{\mathrm {e}}^3}\right )}^2\,\left (8\,\ln \relax (5)-96\,x^4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.56, size = 99, normalized size = 3.54 \begin {gather*} 16 x^{8} - 8 x^{4} \log {\relax (5 )} + 64 x^{2} \log {\left (\log {\left (x + 2 \right )} + e^{e^{3}} \right )}^{3} + 2 x + \left (96 x^{4} - 8 \log {\relax (5 )}\right ) \log {\left (\log {\left (x + 2 \right )} + e^{e^{3}} \right )}^{2} + \left (64 x^{6} - 16 x^{2} \log {\relax (5 )}\right ) \log {\left (\log {\left (x + 2 \right )} + e^{e^{3}} \right )} + 16 \log {\left (\log {\left (x + 2 \right )} + e^{e^{3}} \right )}^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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