Optimal. Leaf size=25 \[ \frac {x}{-3-e^{\frac {4 x}{\log (\log (x))}}-(-1+x)^2} \]
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Rubi [F] time = 6.28, 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 {\left (-4+x^2\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (8-4 x+2 x^2\right ) \log (x) \log ^2(\log (x))+\left (16-16 x+12 x^2-4 x^3+x^4\right ) \log (x) \log ^2(\log (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 {-4 e^{\frac {4 x}{\log (\log (x))}} x+\log (x) \log (\log (x)) \left (4 e^{\frac {4 x}{\log (\log (x))}} x+\left (-4-e^{\frac {4 x}{\log (\log (x))}}+x^2\right ) \log (\log (x))\right )}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))} \, dx\\ &=\int \left (\frac {-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right ) \log (x) \log ^2(\log (x))}-\frac {2 x \left (-8+4 x-2 x^2+8 \log (x) \log (\log (x))-4 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))-x \log (x) \log ^2(\log (x))\right )}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (-8+4 x-2 x^2+8 \log (x) \log (\log (x))-4 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))-x \log (x) \log ^2(\log (x))\right )}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))} \, dx\right )+\int \frac {-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right ) \log (x) \log ^2(\log (x))} \, dx\\ &=-\left (2 \int \frac {x \left (-2 \left (4-2 x+x^2\right )+\log (x) \log (\log (x)) \left (8-4 x+2 x^2+\log (\log (x))-x \log (\log (x))\right )\right )}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))} \, dx\right )+\int \left (-\frac {1}{4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2}-\frac {4 x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right ) \log (x) \log ^2(\log (x))}+\frac {4 x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right ) \log (\log (x))}\right ) \, dx\\ &=-\left (2 \int \left (\frac {x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2}-\frac {x^2}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2}-\frac {8 x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))}+\frac {4 x^2}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))}-\frac {2 x^3}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))}+\frac {8 x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (\log (x))}-\frac {4 x^2}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (\log (x))}+\frac {2 x^3}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (\log (x))}\right ) \, dx\right )-4 \int \frac {x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right ) \log (x) \log ^2(\log (x))} \, dx+4 \int \frac {x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right ) \log (\log (x))} \, dx-\int \frac {1}{4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2} \, dx\\ &=-\left (2 \int \frac {x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2} \, dx\right )+2 \int \frac {x^2}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2} \, dx+4 \int \frac {x^3}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))} \, dx-4 \int \frac {x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right ) \log (x) \log ^2(\log (x))} \, dx-4 \int \frac {x^3}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (\log (x))} \, dx+4 \int \frac {x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right ) \log (\log (x))} \, dx-8 \int \frac {x^2}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))} \, dx+8 \int \frac {x^2}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (\log (x))} \, dx+16 \int \frac {x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (x) \log ^2(\log (x))} \, dx-16 \int \frac {x}{\left (4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2\right )^2 \log (\log (x))} \, dx-\int \frac {1}{4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 23, normalized size = 0.92 \begin {gather*} -\frac {x}{4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 22, normalized size = 0.88 \begin {gather*} -\frac {x}{x^{2} - 2 \, x + e^{\left (\frac {4 \, x}{\log \left (\log \relax (x)\right )}\right )} + 4} \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 [A] time = 0.06, size = 23, normalized size = 0.92
method | result | size |
risch | \(-\frac {x}{{\mathrm e}^{\frac {4 x}{\ln \left (\ln \relax (x )\right )}}+x^{2}-2 x +4}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 22, normalized size = 0.88 \begin {gather*} -\frac {x}{x^{2} - 2 \, x + e^{\left (\frac {4 \, x}{\log \left (\log \relax (x)\right )}\right )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.61, size = 209, normalized size = 8.36 \begin {gather*} -\frac {x\,\left ({\ln \left (\ln \relax (x)\right )}^4\,{\ln \relax (x)}^2+8\,{\ln \left (\ln \relax (x)\right )}^3\,{\ln \relax (x)}^2-8\,{\ln \left (\ln \relax (x)\right )}^2\,\ln \relax (x)\right )-x^2\,\left ({\ln \left (\ln \relax (x)\right )}^4\,{\ln \relax (x)}^2+4\,{\ln \left (\ln \relax (x)\right )}^3\,{\ln \relax (x)}^2-4\,{\ln \left (\ln \relax (x)\right )}^2\,\ln \relax (x)\right )+x^3\,\left (2\,{\ln \left (\ln \relax (x)\right )}^3\,{\ln \relax (x)}^2-2\,{\ln \left (\ln \relax (x)\right )}^2\,\ln \relax (x)\right )}{\left ({\mathrm {e}}^{\frac {4\,x}{\ln \left (\ln \relax (x)\right )}}-2\,x+x^2+4\right )\,\left (2\,x^2\,{\ln \left (\ln \relax (x)\right )}^3\,{\ln \relax (x)}^2-2\,x^2\,{\ln \left (\ln \relax (x)\right )}^2\,\ln \relax (x)-x\,{\ln \left (\ln \relax (x)\right )}^4\,{\ln \relax (x)}^2-4\,x\,{\ln \left (\ln \relax (x)\right )}^3\,{\ln \relax (x)}^2+4\,x\,{\ln \left (\ln \relax (x)\right )}^2\,\ln \relax (x)+{\ln \left (\ln \relax (x)\right )}^4\,{\ln \relax (x)}^2+8\,{\ln \left (\ln \relax (x)\right )}^3\,{\ln \relax (x)}^2-8\,{\ln \left (\ln \relax (x)\right )}^2\,\ln \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 20, normalized size = 0.80 \begin {gather*} - \frac {x}{x^{2} - 2 x + e^{\frac {4 x}{\log {\left (\log {\relax (x )} \right )}}} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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