Optimal. Leaf size=28 \[ 4+(-3+x) \left (2-\frac {2 x}{\frac {e^x}{4+x}+\log (\log (x))}\right )^2 \]
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Rubi [F] time = 17.36, 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 {1536 x+640 x^2-96 x^3-72 x^4-8 x^5+e^x \left (-384-64 x+40 x^2+8 x^3\right )+\left (4 e^{3 x}+e^{2 x} \left (96-112 x-16 x^2+8 x^3\right )+e^x \left (-384 x+288 x^2+144 x^3-20 x^4-8 x^5\right )\right ) \log (x)+\left (-1536-640 x+96 x^2+72 x^3+8 x^4+\left (-1536 x-384 x^2+288 x^3+120 x^4+12 x^5+e^{2 x} (48+12 x)+e^x \left (768-416 x-280 x^2+8 x^4\right )\right ) \log (x)\right ) \log (\log (x))+\left (1536+128 x-480 x^2-168 x^3-16 x^4+e^x \left (192+96 x+12 x^2\right )\right ) \log (x) \log ^2(\log (x))+\left (256+192 x+48 x^2+4 x^3\right ) \log (x) \log ^3(\log (x))}{e^{3 x} \log (x)+e^{2 x} (12+3 x) \log (x) \log (\log (x))+e^x \left (48+24 x+3 x^2\right ) \log (x) \log ^2(\log (x))+\left (64+48 x+12 x^2+x^3\right ) \log (x) \log ^3(\log (x))} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (e^x-x (4+x)+(4+x) \log (\log (x))\right ) \left (2 (-3+x) (4+x)^2+\log (x) \left (e^x \left (24+e^x-24 x-3 x^2+2 x^3\right )+(4+x) \left (24+2 e^x-6 x-3 x^2\right ) \log (\log (x))+(4+x)^2 \log ^2(\log (x))\right )\right )}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx\\ &=4 \int \frac {\left (e^x-x (4+x)+(4+x) \log (\log (x))\right ) \left (2 (-3+x) (4+x)^2+\log (x) \left (e^x \left (24+e^x-24 x-3 x^2+2 x^3\right )+(4+x) \left (24+2 e^x-6 x-3 x^2\right ) \log (\log (x))+(4+x)^2 \log ^2(\log (x))\right )\right )}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx\\ &=4 \int \left (1+\frac {2 \left (12-14 x-2 x^2+x^3\right )}{e^x+4 \log (\log (x))+x \log (\log (x))}+\frac {2 (-3+x) x (4+x)^2 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}-\frac {(4+x) \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2}\right ) \, dx\\ &=4 x-4 \int \frac {(4+x) \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2} \, dx+8 \int \frac {12-14 x-2 x^2+x^3}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx+8 \int \frac {(-3+x) x (4+x)^2 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx\\ &=4 x-4 \int \frac {(4+x) \left (-2 \left (-12+x+x^2\right )+x \log (x) \left (24-24 x-3 x^2+2 x^3+2 \left (-9+x^2\right ) \log (\log (x))\right )\right )}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^2} \, dx+8 \int \left (\frac {12}{e^x+4 \log (\log (x))+x \log (\log (x))}-\frac {14 x}{e^x+4 \log (\log (x))+x \log (\log (x))}-\frac {2 x^2}{e^x+4 \log (\log (x))+x \log (\log (x))}+\frac {x^3}{e^x+4 \log (\log (x))+x \log (\log (x))}\right ) \, dx+8 \int \left (-\frac {48 x \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}-\frac {8 x^2 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}+\frac {5 x^3 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}+\frac {x^4 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3}\right ) \, dx\\ &=4 x-4 \int \left (\frac {4 \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2}+\frac {x \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2}\right ) \, dx+8 \int \frac {x^3}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx+8 \int \frac {x^4 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx-16 \int \frac {x^2}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx+40 \int \frac {x^3 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx-64 \int \frac {x^2 \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx+96 \int \frac {1}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-112 \int \frac {x}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-384 \int \frac {x \left (-4-x+3 x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^3} \, dx\\ &=4 x-4 \int \frac {x \left (24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))\right )}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2} \, dx+8 \int \frac {x^3}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx+8 \int \frac {x^4 (-4-x+x (3+x) \log (x) \log (\log (x)))}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx-16 \int \frac {x^2}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-16 \int \frac {24-2 x-2 x^2+24 x \log (x)-24 x^2 \log (x)-3 x^3 \log (x)+2 x^4 \log (x)-18 x \log (x) \log (\log (x))+2 x^3 \log (x) \log (\log (x))}{\log (x) \left (e^x+4 \log (\log (x))+x \log (\log (x))\right )^2} \, dx+40 \int \frac {x^3 (-4-x+x (3+x) \log (x) \log (\log (x)))}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx-64 \int \frac {x^2 (-4-x+x (3+x) \log (x) \log (\log (x)))}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx+96 \int \frac {1}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-112 \int \frac {x}{e^x+4 \log (\log (x))+x \log (\log (x))} \, dx-384 \int \frac {x (-4-x+x (3+x) \log (x) \log (\log (x)))}{\log (x) \left (e^x+(4+x) \log (\log (x))\right )^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.22, size = 49, normalized size = 1.75 \begin {gather*} 4 x \left (1+\frac {(-3+x) x (4+x)^2}{\left (e^x+(4+x) \log (\log (x))\right )^2}-\frac {2 (-3+x) (4+x)}{e^x+(4+x) \log (\log (x))}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 123, normalized size = 4.39 \begin {gather*} \frac {4 \, {\left (x^{5} + 5 \, x^{4} - 8 \, x^{3} + {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (\log \relax (x)\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + x^{2} - 12 \, x\right )} e^{x} - 2 \, {\left (x^{4} + 5 \, x^{3} - 8 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x} - 48 \, x\right )} \log \left (\log \relax (x)\right )\right )}}{2 \, {\left (x + 4\right )} e^{x} \log \left (\log \relax (x)\right ) + {\left (x^{2} + 8 \, x + 16\right )} \log \left (\log \relax (x)\right )^{2} + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 167, normalized size = 5.96 \begin {gather*} \frac {4 \, {\left (x^{5} - 2 \, x^{4} \log \left (\log \relax (x)\right ) + x^{3} \log \left (\log \relax (x)\right )^{2} + 5 \, x^{4} - 2 \, x^{3} e^{x} - 10 \, x^{3} \log \left (\log \relax (x)\right ) + 2 \, x^{2} e^{x} \log \left (\log \relax (x)\right ) + 8 \, x^{2} \log \left (\log \relax (x)\right )^{2} - 8 \, x^{3} - 2 \, x^{2} e^{x} + 16 \, x^{2} \log \left (\log \relax (x)\right ) + 8 \, x e^{x} \log \left (\log \relax (x)\right ) + 16 \, x \log \left (\log \relax (x)\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} + 24 \, x e^{x} + 96 \, x \log \left (\log \relax (x)\right )\right )}}{x^{2} \log \left (\log \relax (x)\right )^{2} + 2 \, x e^{x} \log \left (\log \relax (x)\right ) + 8 \, x \log \left (\log \relax (x)\right )^{2} + 8 \, e^{x} \log \left (\log \relax (x)\right ) + 16 \, \log \left (\log \relax (x)\right )^{2} + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 64, normalized size = 2.29
method | result | size |
risch | \(4 x +\frac {4 \left (x^{3}-2 x^{2} \ln \left (\ln \relax (x )\right )+x^{2}-2 \,{\mathrm e}^{x} x -2 x \ln \left (\ln \relax (x )\right )-12 x +6 \,{\mathrm e}^{x}+24 \ln \left (\ln \relax (x )\right )\right ) \left (4+x \right ) x}{\left (x \ln \left (\ln \relax (x )\right )+{\mathrm e}^{x}+4 \ln \left (\ln \relax (x )\right )\right )^{2}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 123, normalized size = 4.39 \begin {gather*} \frac {4 \, {\left (x^{5} + 5 \, x^{4} - 8 \, x^{3} + {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (\log \relax (x)\right )^{2} - 48 \, x^{2} + x e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + x^{2} - 12 \, x\right )} e^{x} - 2 \, {\left (x^{4} + 5 \, x^{3} - 8 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x} - 48 \, x\right )} \log \left (\log \relax (x)\right )\right )}}{2 \, {\left (x + 4\right )} e^{x} \log \left (\log \relax (x)\right ) + {\left (x^{2} + 8 \, x + 16\right )} \log \left (\log \relax (x)\right )^{2} + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.70, size = 121, normalized size = 4.32 \begin {gather*} \frac {4\,x\,\left ({\mathrm {e}}^{2\,x}-48\,x+96\,\ln \left (\ln \relax (x)\right )+24\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^x+16\,x\,\ln \left (\ln \relax (x)\right )+8\,x\,{\ln \left (\ln \relax (x)\right )}^2-10\,x^2\,\ln \left (\ln \relax (x)\right )-2\,x^3\,\ln \left (\ln \relax (x)\right )+8\,\ln \left (\ln \relax (x)\right )\,{\mathrm {e}}^x+16\,{\ln \left (\ln \relax (x)\right )}^2-2\,x\,{\mathrm {e}}^x-8\,x^2+5\,x^3+x^4+x^2\,{\ln \left (\ln \relax (x)\right )}^2+2\,x\,\ln \left (\ln \relax (x)\right )\,{\mathrm {e}}^x\right )}{{\left (4\,\ln \left (\ln \relax (x)\right )+{\mathrm {e}}^x+x\,\ln \left (\ln \relax (x)\right )\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.76, size = 131, normalized size = 4.68 \begin {gather*} 4 x + \frac {4 x^{5} - 8 x^{4} \log {\left (\log {\relax (x )} \right )} + 20 x^{4} - 40 x^{3} \log {\left (\log {\relax (x )} \right )} - 32 x^{3} + 64 x^{2} \log {\left (\log {\relax (x )} \right )} - 192 x^{2} + 384 x \log {\left (\log {\relax (x )} \right )} + \left (- 8 x^{3} - 8 x^{2} + 96 x\right ) e^{x}}{x^{2} \log {\left (\log {\relax (x )} \right )}^{2} + 8 x \log {\left (\log {\relax (x )} \right )}^{2} + \left (2 x \log {\left (\log {\relax (x )} \right )} + 8 \log {\left (\log {\relax (x )} \right )}\right ) e^{x} + e^{2 x} + 16 \log {\left (\log {\relax (x )} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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