Optimal. Leaf size=31 \[ \left (2+e^{\frac {x}{\log (\log (4))}}+\frac {x^2}{2 \left (5+x^2\right )-\log (x)}\right )^2 \]
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Rubi [F] time = 4.34, 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 {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{\left (-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)\right ) \log (\log (4))} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)} \, dx}{\log (\log (4))}\\ &=\frac {\int \frac {2 \left (5 \left (4+x^2\right )+2 e^{\frac {x}{\log (\log (4))}} \left (5+x^2\right )-\left (2+e^{\frac {x}{\log (\log (4))}}\right ) \log (x)\right ) \left (4 e^{\frac {x}{\log (\log (4))}} \left (5+x^2\right )^2+e^{\frac {x}{\log (\log (4))}} \log ^2(x)+21 x \log (\log (4))-2 \log (x) \left (2 e^{\frac {x}{\log (\log (4))}} \left (5+x^2\right )+x \log (\log (4))\right )\right )}{\left (2 \left (5+x^2\right )-\log (x)\right )^3} \, dx}{\log (\log (4))}\\ &=\frac {2 \int \frac {\left (5 \left (4+x^2\right )+2 e^{\frac {x}{\log (\log (4))}} \left (5+x^2\right )-\left (2+e^{\frac {x}{\log (\log (4))}}\right ) \log (x)\right ) \left (4 e^{\frac {x}{\log (\log (4))}} \left (5+x^2\right )^2+e^{\frac {x}{\log (\log (4))}} \log ^2(x)+21 x \log (\log (4))-2 \log (x) \left (2 e^{\frac {x}{\log (\log (4))}} \left (5+x^2\right )+x \log (\log (4))\right )\right )}{\left (2 \left (5+x^2\right )-\log (x)\right )^3} \, dx}{\log (\log (4))}\\ &=\frac {2 \int \left (e^{\frac {2 x}{\log (\log (4))}}+\frac {105 x \left (4+x^2\right ) \log (\log (4))}{\left (10+2 x^2-\log (x)\right )^3}-\frac {42 x \log (x) \log (\log (4))}{\left (10+2 x^2-\log (x)\right )^3}-\frac {10 x \left (4+x^2\right ) \log (x) \log (\log (4))}{\left (10+2 x^2-\log (x)\right )^3}+\frac {4 x \log ^2(x) \log (\log (4))}{\left (10+2 x^2-\log (x)\right )^3}+\frac {e^{\frac {x}{\log (\log (4))}} \left (200+90 x^2+10 x^4-40 \log (x)-9 x^2 \log (x)+2 \log ^2(x)+21 x \log (\log (4))-2 x \log (x) \log (\log (4))\right )}{\left (10+2 x^2-\log (x)\right )^2}\right ) \, dx}{\log (\log (4))}\\ &=8 \int \frac {x \log ^2(x)}{\left (10+2 x^2-\log (x)\right )^3} \, dx-20 \int \frac {x \left (4+x^2\right ) \log (x)}{\left (10+2 x^2-\log (x)\right )^3} \, dx-84 \int \frac {x \log (x)}{\left (10+2 x^2-\log (x)\right )^3} \, dx+210 \int \frac {x \left (4+x^2\right )}{\left (10+2 x^2-\log (x)\right )^3} \, dx+\frac {2 \int e^{\frac {2 x}{\log (\log (4))}} \, dx}{\log (\log (4))}+\frac {2 \int \frac {e^{\frac {x}{\log (\log (4))}} \left (200+90 x^2+10 x^4-40 \log (x)-9 x^2 \log (x)+2 \log ^2(x)+21 x \log (\log (4))-2 x \log (x) \log (\log (4))\right )}{\left (10+2 x^2-\log (x)\right )^2} \, dx}{\log (\log (4))}\\ &=e^{\frac {2 x}{\log (\log (4))}}+\frac {2 e^{\frac {x}{\log (\log (4))}} \left (200+90 x^2+10 x^4-40 \log (x)-9 x^2 \log (x)+2 \log ^2(x)\right )}{\left (10+2 x^2-\log (x)\right )^2}+8 \int \left (\frac {4 x \left (5+x^2\right )^2}{\left (10+2 x^2-\log (x)\right )^3}-\frac {4 x \left (5+x^2\right )}{\left (10+2 x^2-\log (x)\right )^2}+\frac {x}{10+2 x^2-\log (x)}\right ) \, dx-20 \int \left (\frac {2 x \left (20+9 x^2+x^4\right )}{\left (10+2 x^2-\log (x)\right )^3}-\frac {x \left (4+x^2\right )}{\left (10+2 x^2-\log (x)\right )^2}\right ) \, dx-84 \int \left (\frac {2 x \left (5+x^2\right )}{\left (10+2 x^2-\log (x)\right )^3}-\frac {x}{\left (10+2 x^2-\log (x)\right )^2}\right ) \, dx+210 \int \left (\frac {4 x}{\left (10+2 x^2-\log (x)\right )^3}+\frac {x^3}{\left (10+2 x^2-\log (x)\right )^3}\right ) \, dx\\ &=e^{\frac {2 x}{\log (\log (4))}}+\frac {2 e^{\frac {x}{\log (\log (4))}} \left (200+90 x^2+10 x^4-40 \log (x)-9 x^2 \log (x)+2 \log ^2(x)\right )}{\left (10+2 x^2-\log (x)\right )^2}+8 \int \frac {x}{10+2 x^2-\log (x)} \, dx+20 \int \frac {x \left (4+x^2\right )}{\left (10+2 x^2-\log (x)\right )^2} \, dx+32 \int \frac {x \left (5+x^2\right )^2}{\left (10+2 x^2-\log (x)\right )^3} \, dx-32 \int \frac {x \left (5+x^2\right )}{\left (10+2 x^2-\log (x)\right )^2} \, dx-40 \int \frac {x \left (20+9 x^2+x^4\right )}{\left (10+2 x^2-\log (x)\right )^3} \, dx+84 \int \frac {x}{\left (10+2 x^2-\log (x)\right )^2} \, dx-168 \int \frac {x \left (5+x^2\right )}{\left (10+2 x^2-\log (x)\right )^3} \, dx+210 \int \frac {x^3}{\left (10+2 x^2-\log (x)\right )^3} \, dx+840 \int \frac {x}{\left (10+2 x^2-\log (x)\right )^3} \, dx\\ &=e^{\frac {2 x}{\log (\log (4))}}+\frac {2 e^{\frac {x}{\log (\log (4))}} \left (200+90 x^2+10 x^4-40 \log (x)-9 x^2 \log (x)+2 \log ^2(x)\right )}{\left (10+2 x^2-\log (x)\right )^2}+8 \int \frac {x}{10+2 x^2-\log (x)} \, dx+20 \int \left (\frac {4 x}{\left (10+2 x^2-\log (x)\right )^2}+\frac {x^3}{\left (10+2 x^2-\log (x)\right )^2}\right ) \, dx+32 \int \left (\frac {25 x}{\left (10+2 x^2-\log (x)\right )^3}+\frac {10 x^3}{\left (10+2 x^2-\log (x)\right )^3}+\frac {x^5}{\left (10+2 x^2-\log (x)\right )^3}\right ) \, dx-32 \int \left (\frac {5 x}{\left (10+2 x^2-\log (x)\right )^2}+\frac {x^3}{\left (10+2 x^2-\log (x)\right )^2}\right ) \, dx-40 \int \left (\frac {20 x}{\left (10+2 x^2-\log (x)\right )^3}+\frac {9 x^3}{\left (10+2 x^2-\log (x)\right )^3}+\frac {x^5}{\left (10+2 x^2-\log (x)\right )^3}\right ) \, dx+84 \int \frac {x}{\left (10+2 x^2-\log (x)\right )^2} \, dx-168 \int \left (\frac {5 x}{\left (10+2 x^2-\log (x)\right )^3}+\frac {x^3}{\left (10+2 x^2-\log (x)\right )^3}\right ) \, dx+210 \int \frac {x^3}{\left (10+2 x^2-\log (x)\right )^3} \, dx+840 \int \frac {x}{\left (10+2 x^2-\log (x)\right )^3} \, dx\\ &=e^{\frac {2 x}{\log (\log (4))}}+\frac {2 e^{\frac {x}{\log (\log (4))}} \left (200+90 x^2+10 x^4-40 \log (x)-9 x^2 \log (x)+2 \log ^2(x)\right )}{\left (10+2 x^2-\log (x)\right )^2}+8 \int \frac {x}{10+2 x^2-\log (x)} \, dx+20 \int \frac {x^3}{\left (10+2 x^2-\log (x)\right )^2} \, dx+32 \int \frac {x^5}{\left (10+2 x^2-\log (x)\right )^3} \, dx-32 \int \frac {x^3}{\left (10+2 x^2-\log (x)\right )^2} \, dx-40 \int \frac {x^5}{\left (10+2 x^2-\log (x)\right )^3} \, dx+80 \int \frac {x}{\left (10+2 x^2-\log (x)\right )^2} \, dx+84 \int \frac {x}{\left (10+2 x^2-\log (x)\right )^2} \, dx-160 \int \frac {x}{\left (10+2 x^2-\log (x)\right )^2} \, dx-168 \int \frac {x^3}{\left (10+2 x^2-\log (x)\right )^3} \, dx+210 \int \frac {x^3}{\left (10+2 x^2-\log (x)\right )^3} \, dx+320 \int \frac {x^3}{\left (10+2 x^2-\log (x)\right )^3} \, dx-360 \int \frac {x^3}{\left (10+2 x^2-\log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.17, size = 68, normalized size = 2.19 \begin {gather*} 4 e^{\frac {x}{\log (\log (4))}}+e^{\frac {2 x}{\log (\log (4))}}+\frac {2 \left (2+e^{\frac {x}{\log (\log (4))}}\right ) x^2}{2 \left (5+x^2\right )-\log (x)}+\frac {x^4}{\left (-2 \left (5+x^2\right )+\log (x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 124, normalized size = 4.00 \begin {gather*} \frac {9 \, x^{4} - 4 \, x^{2} \log \relax (x) + 40 \, x^{2} + {\left (4 \, x^{4} + 40 \, x^{2} - 4 \, {\left (x^{2} + 5\right )} \log \relax (x) + \log \relax (x)^{2} + 100\right )} e^{\left (\frac {2 \, x}{\log \left (2 \, \log \relax (2)\right )}\right )} + 2 \, {\left (10 \, x^{4} + 90 \, x^{2} - {\left (9 \, x^{2} + 40\right )} \log \relax (x) + 2 \, \log \relax (x)^{2} + 200\right )} e^{\left (\frac {x}{\log \left (2 \, \log \relax (2)\right )}\right )}}{4 \, x^{4} + 40 \, x^{2} - 4 \, {\left (x^{2} + 5\right )} \log \relax (x) + \log \relax (x)^{2} + 100} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 262, normalized size = 8.45
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {2 x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}} \ln \relax (2)}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}+\frac {{\mathrm e}^{\frac {2 x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}} \ln \left (\ln \relax (2)\right )}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}+\frac {4 \ln \relax (2) {\mathrm e}^{\frac {x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}}}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}+\frac {4 \ln \left (\ln \relax (2)\right ) {\mathrm e}^{\frac {x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}}}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}+\frac {\left (4 \ln \relax (2) x^{2} {\mathrm e}^{\frac {x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}}+4 \ln \left (\ln \relax (2)\right ) x^{2} {\mathrm e}^{\frac {x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}}+9 x^{2} \ln \relax (2)-2 \ln \relax (2) {\mathrm e}^{\frac {x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}} \ln \relax (x )+9 x^{2} \ln \left (\ln \relax (2)\right )-2 \ln \left (\ln \relax (2)\right ) {\mathrm e}^{\frac {x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}} \ln \relax (x )+20 \ln \relax (2) {\mathrm e}^{\frac {x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}}-4 \ln \relax (2) \ln \relax (x )+20 \ln \left (\ln \relax (2)\right ) {\mathrm e}^{\frac {x}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}}-4 \ln \relax (x ) \ln \left (\ln \relax (2)\right )+40 \ln \relax (2)+40 \ln \left (\ln \relax (2)\right )\right ) x^{2}}{\left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right ) \left (2 x^{2}+10-\ln \relax (x )\right )^{2}}\) | \(262\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 233, normalized size = 7.52 \begin {gather*} \frac {9 \, x^{4} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} - 4 \, x^{2} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} \log \relax (x) + 40 \, x^{2} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + {\left (4 \, x^{4} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + 40 \, x^{2} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} \log \relax (x)^{2} - 4 \, {\left (x^{2} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + 5 \, \log \relax (2) + 5 \, \log \left (\log \relax (2)\right )\right )} \log \relax (x) + 100 \, \log \relax (2) + 100 \, \log \left (\log \relax (2)\right )\right )} e^{\left (\frac {2 \, x}{\log \relax (2) + \log \left (\log \relax (2)\right )}\right )} + 2 \, {\left (10 \, x^{4} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + 90 \, x^{2} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + 2 \, {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} \log \relax (x)^{2} - {\left (9 \, x^{2} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + 40 \, \log \relax (2) + 40 \, \log \left (\log \relax (2)\right )\right )} \log \relax (x) + 200 \, \log \relax (2) + 200 \, \log \left (\log \relax (2)\right )\right )} e^{\left (\frac {x}{\log \relax (2) + \log \left (\log \relax (2)\right )}\right )}}{{\left (4 \, x^{4} + 40 \, x^{2} - 4 \, {\left (x^{2} + 5\right )} \log \relax (x) + \log \relax (x)^{2} + 100\right )} \log \left (2 \, \log \relax (2)\right )} \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}}^{\frac {2\,x}{\ln \left (2\,\ln \relax (2)\right )}}\,\left ({\ln \relax (x)}^2\,\left (12\,x^2+60\right )-\ln \relax (x)\,\left (24\,x^4+240\,x^2+600\right )-2\,{\ln \relax (x)}^3+1200\,x^2+240\,x^4+16\,x^6+2000\right )+\ln \left (2\,\ln \relax (2)\right )\,\left (840\,x+8\,x\,{\ln \relax (x)}^2-\ln \relax (x)\,\left (20\,x^3+164\,x\right )+210\,x^3\right )+{\mathrm {e}}^{\frac {x}{\ln \left (2\,\ln \relax (2)\right )}}\,\left ({\ln \relax (x)}^2\,\left (26\,x^2+120\right )-\ln \relax (x)\,\left (56\,x^4+520\,x^2+1200\right )-4\,{\ln \relax (x)}^3+\ln \left (2\,\ln \relax (2)\right )\,\left (420\,x+4\,x\,{\ln \relax (x)}^2-\ln \relax (x)\,\left (8\,x^3+82\,x\right )+84\,x^3\right )+2600\,x^2+560\,x^4+40\,x^6+4000\right )}{\ln \left (2\,\ln \relax (2)\right )\,\left ({\ln \relax (x)}^2\,\left (6\,x^2+30\right )-\ln \relax (x)\,\left (12\,x^4+120\,x^2+300\right )-{\ln \relax (x)}^3+600\,x^2+120\,x^4+8\,x^6+1000\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.80, size = 100, normalized size = 3.23 \begin {gather*} \frac {\left (2 x^{2} - \log {\relax (x )} + 10\right ) e^{\frac {2 x}{\log {\left (2 \log {\relax (2 )} \right )}}} + \left (10 x^{2} - 4 \log {\relax (x )} + 40\right ) e^{\frac {x}{\log {\left (2 \log {\relax (2 )} \right )}}}}{2 x^{2} - \log {\relax (x )} + 10} + \frac {9 x^{4} - 4 x^{2} \log {\relax (x )} + 40 x^{2}}{4 x^{4} + 40 x^{2} + \left (- 4 x^{2} - 20\right ) \log {\relax (x )} + \log {\relax (x )}^{2} + 100} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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