Optimal. Leaf size=20 \[ e^{\sqrt [3]{2}}+\frac {1}{4} (x \log (x))^{x^2} \]
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Rubi [F] time = 0.38, 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 {(x \log (x))^{x^2} (x+x \log (x)+2 x \log (x) \log (x \log (x)))}{4 \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {(x \log (x))^{x^2} (x+x \log (x)+2 x \log (x) \log (x \log (x)))}{\log (x)} \, dx\\ &=\frac {1}{4} \int \left (\frac {x (x \log (x))^{x^2} (1+\log (x))}{\log (x)}+2 x (x \log (x))^{x^2} \log (x \log (x))\right ) \, dx\\ &=\frac {1}{4} \int \frac {x (x \log (x))^{x^2} (1+\log (x))}{\log (x)} \, dx+\frac {1}{2} \int x (x \log (x))^{x^2} \log (x \log (x)) \, dx\\ &=\frac {1}{4} \int \left (x (x \log (x))^{x^2}+\frac {x (x \log (x))^{x^2}}{\log (x)}\right ) \, dx+\frac {1}{2} \int x (x \log (x))^{x^2} \log (x \log (x)) \, dx\\ &=\frac {1}{4} \int x (x \log (x))^{x^2} \, dx+\frac {1}{4} \int \frac {x (x \log (x))^{x^2}}{\log (x)} \, dx+\frac {1}{2} \int x (x \log (x))^{x^2} \log (x \log (x)) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 12, normalized size = 0.60 \begin {gather*} \frac {1}{4} (x \log (x))^{x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (x^{2} \log \left (x \log \relax (x)\right ) - 2 \, \log \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (x^{2} \log \left (x \log \relax (x)\right ) - 2 \, \log \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 91, normalized size = 4.55
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {x^{2} \left (-i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )-i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )+2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\right )}{2}}}{4}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 17, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, e^{\left (x^{2} \log \relax (x) + x^{2} \log \left (\log \relax (x)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 10, normalized size = 0.50 \begin {gather*} \frac {{\left (x\,\ln \relax (x)\right )}^{x^2}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 12, normalized size = 0.60 \begin {gather*} \frac {e^{x^{2} \log {\left (x \log {\relax (x )} \right )}}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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