Optimal. Leaf size=29 \[ e \left (-x+x^2 \log \left (\frac {2}{x}-x-x (2+\log (x))^2\right )\right ) \]
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Rubi [F] time = 0.94, 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 \left (2+2 x-5 x^2+9 x^3\right )+e \left (-4 x^2+6 x^3\right ) \log (x)+e \left (-x^2+x^3\right ) \log ^2(x)+\left (e \left (-4 x+10 x^3\right )+8 e x^3 \log (x)+2 e x^3 \log ^2(x)\right ) \log \left (\frac {2-5 x^2-4 x^2 \log (x)-x^2 \log ^2(x)}{x}\right )}{-2+5 x^2+4 x^2 \log (x)+x^2 \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 \left (\frac {e \left (2+2 x-5 x^2+9 x^3-4 x^2 \log (x)+6 x^3 \log (x)-x^2 \log ^2(x)+x^3 \log ^2(x)\right )}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)}+2 e x \log \left (\frac {2}{x}-5 x-4 x \log (x)-x \log ^2(x)\right )\right ) \, dx\\ &=e \int \frac {2+2 x-5 x^2+9 x^3-4 x^2 \log (x)+6 x^3 \log (x)-x^2 \log ^2(x)+x^3 \log ^2(x)}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)} \, dx+(2 e) \int x \log \left (\frac {2}{x}-5 x-4 x \log (x)-x \log ^2(x)\right ) \, dx\\ &=e \int \left (-1+x+\frac {2 x \left (2+2 x^2+x^2 \log (x)\right )}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)}\right ) \, dx+(2 e) \int x \log \left (\frac {2}{x}-5 x-4 x \log (x)-x \log ^2(x)\right ) \, dx\\ &=-e x+\frac {e x^2}{2}+(2 e) \int \frac {x \left (2+2 x^2+x^2 \log (x)\right )}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)} \, dx+(2 e) \int x \log \left (\frac {2}{x}-5 x-4 x \log (x)-x \log ^2(x)\right ) \, dx\\ &=-e x+\frac {e x^2}{2}+(2 e) \int \left (\frac {2 x}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)}+\frac {2 x^3}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)}+\frac {x^3 \log (x)}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)}\right ) \, dx+(2 e) \int x \log \left (\frac {2}{x}-5 x-4 x \log (x)-x \log ^2(x)\right ) \, dx\\ &=-e x+\frac {e x^2}{2}+(2 e) \int \frac {x^3 \log (x)}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)} \, dx+(2 e) \int x \log \left (\frac {2}{x}-5 x-4 x \log (x)-x \log ^2(x)\right ) \, dx+(4 e) \int \frac {x}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)} \, dx+(4 e) \int \frac {x^3}{-2+5 x^2+4 x^2 \log (x)+x^2 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 32, normalized size = 1.10 \begin {gather*} e \left (-x+x^2 \log \left (\frac {2}{x}-5 x-4 x \log (x)-x \log ^2(x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 40, normalized size = 1.38 \begin {gather*} x^{2} e \log \left (-\frac {x^{2} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x) + 5 \, x^{2} - 2}{x}\right ) - x e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 45, normalized size = 1.55 \begin {gather*} x^{2} e \log \left (-x^{2} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (x) - 5 \, x^{2} + 2\right ) - x^{2} e \log \relax (x) - x e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 270, normalized size = 9.31
method | result | size |
risch | \(x^{2} {\mathrm e} \ln \left (-2+\left (\ln \relax (x )^{2}+4 \ln \relax (x )+5\right ) x^{2}\right )-x^{2} {\mathrm e} \ln \relax (x )+\frac {i \pi \,{\mathrm e} x^{2} \mathrm {csgn}\left (i \left (-2+\left (\ln \relax (x )^{2}+4 \ln \relax (x )+5\right ) x^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-2+\left (\ln \relax (x )^{2}+4 \ln \relax (x )+5\right ) x^{2}\right )}{x}\right )^{2}}{2}-\frac {i \pi \,{\mathrm e} x^{2} \mathrm {csgn}\left (i \left (-2+\left (\ln \relax (x )^{2}+4 \ln \relax (x )+5\right ) x^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-2+\left (\ln \relax (x )^{2}+4 \ln \relax (x )+5\right ) x^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )}{2}+\frac {i \pi \,{\mathrm e} x^{2} \mathrm {csgn}\left (\frac {i \left (-2+\left (\ln \relax (x )^{2}+4 \ln \relax (x )+5\right ) x^{2}\right )}{x}\right )^{3}}{2}-i \pi \,{\mathrm e} x^{2} \mathrm {csgn}\left (\frac {i \left (-2+\left (\ln \relax (x )^{2}+4 \ln \relax (x )+5\right ) x^{2}\right )}{x}\right )^{2}+\frac {i \pi \,{\mathrm e} x^{2} \mathrm {csgn}\left (\frac {i \left (-2+\left (\ln \relax (x )^{2}+4 \ln \relax (x )+5\right ) x^{2}\right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )}{2}+i {\mathrm e} \pi \,x^{2}-x \,{\mathrm e}\) | \(270\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 45, normalized size = 1.55 \begin {gather*} x^{2} e \log \left (-x^{2} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (x) - 5 \, x^{2} + 2\right ) - x^{2} e \log \relax (x) - x e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.04, size = 40, normalized size = 1.38 \begin {gather*} x^2\,\mathrm {e}\,\ln \left (-\frac {x^2\,{\ln \relax (x)}^2+4\,x^2\,\ln \relax (x)+5\,x^2-2}{x}\right )-x\,\mathrm {e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 37, normalized size = 1.28 \begin {gather*} e x^{2} \log {\left (\frac {- x^{2} \log {\relax (x )}^{2} - 4 x^{2} \log {\relax (x )} - 5 x^{2} + 2}{x} \right )} - e x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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