Optimal. Leaf size=28 \[ \frac {4}{x \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \]
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Rubi [F]
time = 10.14, 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 {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\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 {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx\\ &=\int \left (-\frac {4 \left (x^2+\log (x)+x^2 \log (x)+\log ^2(x)+2 \log (x) \log \left (x^2+\log (x)\right )+4 x^2 \log (x) \log \left (x^2+\log (x)\right )\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}-\frac {4 \left (-2 x^2 \log (x)-2 \log ^2(x)+x^2 \log \left (x^2\right )+\log (x) \log \left (x^2\right )+2 \log (x) \log \left (x^2\right ) \log \left (x^2+\log (x)\right )+4 x^2 \log (x) \log \left (x^2\right ) \log \left (x^2+\log (x)\right )\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {x^2+\log (x)+x^2 \log (x)+\log ^2(x)+2 \log (x) \log \left (x^2+\log (x)\right )+4 x^2 \log (x) \log \left (x^2+\log (x)\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx\right )-4 \int \frac {-2 x^2 \log (x)-2 \log ^2(x)+x^2 \log \left (x^2\right )+\log (x) \log \left (x^2\right )+2 \log (x) \log \left (x^2\right ) \log \left (x^2+\log (x)\right )+4 x^2 \log (x) \log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx\\ &=-\left (4 \int \left (\frac {1}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {1}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {1}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {\log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {4 \log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}+\frac {2 \log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )}\right ) \, dx\right )-4 \int \frac {-2 \log ^2(x)+x^2 \log \left (x^2\right )+\log (x) \left (-2 x^2+\log \left (x^2\right ) \left (1+\left (2+4 x^2\right ) \log \left (x^2+\log (x)\right )\right )\right )}{x^2 \log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx\\ &=-\left (4 \int \frac {1}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx\right )-4 \int \frac {1}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {1}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {\log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \left (-\frac {2}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}-\frac {2 \log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}+\frac {\log \left (x^2\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}+\frac {\log \left (x^2\right )}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}+\frac {4 \log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}+\frac {2 \log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}\right ) \, dx-8 \int \frac {\log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-16 \int \frac {\log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx\\ &=-\left (4 \int \frac {1}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx\right )-4 \int \frac {1}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {1}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {\log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx-4 \int \frac {\log \left (x^2\right )}{\log (x) \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx+8 \int \frac {1}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx+8 \int \frac {\log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx-8 \int \frac {\log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-8 \int \frac {\log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx-16 \int \frac {\log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \, dx-16 \int \frac {\log \left (x^2\right ) \log \left (x^2+\log (x)\right )}{\left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 28, normalized size = 1.00 \begin {gather*} \frac {4}{x \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.28, size = 74, normalized size = 2.64
method | result | size |
risch | \(-\frac {8 i}{x \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \ln \left (x \right ) {\mathrm e}^{\ln \left (\ln \left (x \right )+x^{2}\right )^{2}}+4 i \ln \left (x \right )\right )}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 24, normalized size = 0.86 \begin {gather*} \frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 24, normalized size = 0.86 \begin {gather*} \frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 24, normalized size = 0.86 \begin {gather*} \frac {4}{x e^{\log {\left (x^{2} + \log {\left (x \right )} \right )}^{2}} \log {\left (x \right )} - 2 x \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 24, normalized size = 0.86 \begin {gather*} \frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.07, size = 310, normalized size = 11.07 \begin {gather*} \frac {4\,\left (2\,x^2\,\ln \left (x\right )+2\,{\ln \left (x\right )}^2\right )\,{\left (x\,\ln \left (x\right )+x^3\right )}^2-4\,\ln \left (x^2\right )\,{\left (x\,\ln \left (x\right )+x^3\right )}^2\,\left (\ln \left (x\right )+2\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )+x^2+4\,x^2\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\right )}{x^2\,\left (\ln \left (x\right )+x^2\right )\,\left (\ln \left (x^2\right )-{\mathrm {e}}^{{\ln \left (\ln \left (x\right )+x^2\right )}^2}\,\ln \left (x\right )\right )\,\left (x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+x\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2+8\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^3+8\,x^5\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2+4\,x\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^3+2\,x\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^5\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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