Optimal. Leaf size=30 \[ \frac {e \left (x+e^{4/x} x\right )}{2 \log \left (4-\frac {\log ^2(x)}{x}\right )} \]
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Rubi [F] time = 3.96, 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 (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\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 {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{x \left (-8 x+2 \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx\\ &=\int \frac {-\left (\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)\right )-\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)-\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{2 x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {1}{2} \int \frac {-\left (\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)\right )-\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)-\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {1}{2} \int \left (\frac {e \left (2 \log (x)-\log ^2(x)+4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{\left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {e^{1+\frac {4}{x}} \left (2 x \log (x)-x \log ^2(x)-16 x \log \left (4-\frac {\log ^2(x)}{x}\right )+4 x^2 \log \left (4-\frac {\log ^2(x)}{x}\right )+4 \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )-x \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{1+\frac {4}{x}} \left (2 x \log (x)-x \log ^2(x)-16 x \log \left (4-\frac {\log ^2(x)}{x}\right )+4 x^2 \log \left (4-\frac {\log ^2(x)}{x}\right )+4 \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )-x \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx+\frac {1}{2} e \int \frac {2 \log (x)-\log ^2(x)+4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )}{\left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {e^{1+\frac {4}{x}} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{2 \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{2} e \int \left (\frac {(-2+\log (x)) \log (x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )}\right ) \, dx\\ &=\frac {e^{1+\frac {4}{x}} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{2 \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{2} e \int \frac {(-2+\log (x)) \log (x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx+\frac {1}{2} e \int \frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {e^{1+\frac {4}{x}} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{2 \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{2} e \int \left (-\frac {2 \log (x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {\log ^2(x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}\right ) \, dx+\frac {1}{2} e \int \frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {e^{1+\frac {4}{x}} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{2 \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{2} e \int \frac {\log ^2(x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx+\frac {1}{2} e \int \frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )} \, dx-e \int \frac {\log (x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 29, normalized size = 0.97 \begin {gather*} \frac {e \left (1+e^{4/x}\right ) x}{2 \log \left (4-\frac {\log ^2(x)}{x}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 33, normalized size = 1.10 \begin {gather*} \frac {x e + x e^{\left (\frac {x + 4}{x}\right )}}{2 \, \log \left (-\frac {\log \relax (x)^{2} - 4 \, x}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 35, normalized size = 1.17 \begin {gather*} \frac {x e + x e^{\left (\frac {x + 4}{x}\right )}}{2 \, {\left (\log \left (-\log \relax (x)^{2} + 4 \, x\right ) - \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.45, size = 166, normalized size = 5.53
method | result | size |
risch | \(-\frac {i \left ({\mathrm e}^{\frac {4}{x}}+1\right ) x \,{\mathrm e}}{-\pi \,\mathrm {csgn}\left (i \left (\frac {\ln \relax (x )^{2}}{4}-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \relax (x )^{2}}{4}-x \right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\frac {\ln \relax (x )^{2}}{4}-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \relax (x )^{2}}{4}-x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+\pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \relax (x )^{2}}{4}-x \right )}{x}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \relax (x )^{2}}{4}-x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-4 i \ln \relax (2)+2 i \ln \relax (x )-2 i \ln \left (-\frac {\ln \relax (x )^{2}}{4}+x \right )}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 35, normalized size = 1.17 \begin {gather*} \frac {x e + x e^{\left (\frac {4}{x} + 1\right )}}{2 \, {\left (\log \left (-\log \relax (x)^{2} + 4 \, x\right ) - \log \relax (x)\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 {{\ln \relax (x)}^2\,\left (x\,\mathrm {e}+x\,\mathrm {e}\,{\mathrm {e}}^{4/x}\right )+\ln \left (\frac {4\,x-{\ln \relax (x)}^2}{x}\right )\,\left ({\ln \relax (x)}^2\,\left (x\,\mathrm {e}+\mathrm {e}\,{\mathrm {e}}^{4/x}\,\left (x-4\right )\right )-4\,x^2\,\mathrm {e}+\mathrm {e}\,{\mathrm {e}}^{4/x}\,\left (16\,x-4\,x^2\right )\right )-\ln \relax (x)\,\left (2\,x\,\mathrm {e}+2\,x\,\mathrm {e}\,{\mathrm {e}}^{4/x}\right )}{{\ln \left (\frac {4\,x-{\ln \relax (x)}^2}{x}\right )}^2\,\left (2\,x\,{\ln \relax (x)}^2-8\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 41, normalized size = 1.37 \begin {gather*} \frac {e x e^{\frac {4}{x}}}{2 \log {\left (\frac {4 x - \log {\relax (x )}^{2}}{x} \right )}} + \frac {e x}{2 \log {\left (\frac {4 x - \log {\relax (x )}^{2}}{x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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