Optimal. Leaf size=33 \[ x+\frac {1}{5} \left (3+\frac {x \log (x)}{\log \left (\frac {2-\frac {x}{\log \left (x^2\right )}}{5 x}\right )}\right ) \]
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Rubi [F] time = 2.18, 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 {-2 x \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \log (x)-2 \log (x) \log ^2\left (x^2\right )-\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )-\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{5 \left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx\\ &=\frac {1}{5} \int \frac {2 x \log (x)-2 \log (x) \log ^2\left (x^2\right )-\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )-\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx\\ &=\frac {1}{5} \int \left (5+\frac {2 \log (x) \left (x-\log ^2\left (x^2\right )\right )}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}+\frac {1+\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \, dx\\ &=x+\frac {1}{5} \int \frac {1+\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx+\frac {2}{5} \int \frac {\log (x) \left (x-\log ^2\left (x^2\right )\right )}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx\\ &=x+\frac {1}{5} \int \left (\frac {1}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}+\frac {\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \, dx+\frac {2}{5} \int \left (\frac {x \log (x)}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}-\frac {\log (x) \log \left (x^2\right )}{\left (x-2 \log \left (x^2\right )\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \, dx\\ &=x+\frac {1}{5} \int \frac {1}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx+\frac {1}{5} \int \frac {\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx+\frac {2}{5} \int \frac {x \log (x)}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx-\frac {2}{5} \int \frac {\log (x) \log \left (x^2\right )}{\left (x-2 \log \left (x^2\right )\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 32, normalized size = 0.97 \begin {gather*} -\frac {1}{5} x \left (-5-\frac {\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 44, normalized size = 1.33 \begin {gather*} \frac {x \log \relax (x) + 5 \, x \log \left (-\frac {x - 4 \, \log \relax (x)}{10 \, x \log \relax (x)}\right )}{5 \, \log \left (-\frac {x - 4 \, \log \relax (x)}{10 \, x \log \relax (x)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.18, size = 218, normalized size = 6.61 \begin {gather*} x - \frac {4 \, x^{2} \log \left (x^{2}\right ) \log \relax (x)^{2} - 8 \, x \log \left (x^{2}\right )^{2} \log \relax (x)^{2} - x^{3} \log \left (x^{2}\right ) + 2 \, x^{2} \log \left (x^{2}\right )^{2}}{10 \, {\left (x \log \left (x^{2}\right )^{2} \log \relax (x) - 4 \, \log \left (x^{2}\right )^{2} \log \relax (x)^{2} - x \log \left (x^{2}\right )^{2} \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) + 4 \, \log \left (x^{2}\right )^{2} \log \relax (x) \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) + x \log \left (x^{2}\right )^{2} \log \left (5 \, \log \left (x^{2}\right )\right ) - 4 \, \log \left (x^{2}\right )^{2} \log \relax (x) \log \left (5 \, \log \left (x^{2}\right )\right ) - x^{2} \log \relax (x) + 4 \, x \log \relax (x)^{2} + x^{2} \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) - 4 \, x \log \relax (x) \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) - x^{2} \log \left (5 \, \log \left (x^{2}\right )\right ) + 4 \, x \log \relax (x) \log \left (5 \, \log \left (x^{2}\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.65, size = 1531, normalized size = 46.39
method | result | size |
risch | \(x -\frac {2 x \ln \relax (x )}{5 \left (-2 \ln \relax (2)+2 \ln \relax (5)+2 \ln \relax (x )+2 \ln \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )-2 \ln \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right ) \mathrm {csgn}\left (i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{x \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )}\right )^{3}-i \pi \,\mathrm {csgn}\left (i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{x \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{x \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{x \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right )^{2}\right )}\) | \(1531\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 57, normalized size = 1.73 \begin {gather*} \frac {5 \, x {\left (\log \relax (5) + \log \relax (2)\right )} + 4 \, x \log \relax (x) - 5 \, x \log \left (-x + 4 \, \log \relax (x)\right ) + 5 \, x \log \left (\log \relax (x)\right )}{5 \, {\left (\log \relax (5) + \log \relax (2) + \log \relax (x) - \log \left (-x + 4 \, \log \relax (x)\right ) + \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 = 4.02, size = 296, normalized size = 8.97 \begin {gather*} \frac {4\,x}{5}+\frac {\frac {x\,\ln \relax (x)}{5}-\frac {x\,\ln \left (-\frac {\frac {x}{5}-\frac {2\,\ln \left (x^2\right )}{5}}{x\,\ln \left (x^2\right )}\right )\,\ln \left (x^2\right )\,\left (\ln \relax (x)+1\right )\,\left (x-2\,\ln \left (x^2\right )\right )}{10\,\left (4\,\ln \relax (x)\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-x+4\,{\ln \relax (x)}^2+{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2\right )}}{\ln \left (-\frac {\frac {x}{5}-\frac {2\,\ln \left (x^2\right )}{5}}{x\,\ln \left (x^2\right )}\right )}-\frac {x\,\ln \relax (x)}{5}+\frac {x^2}{20}-\frac {\frac {2\,x^5\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-32\,x^4\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )+16\,x^4\,{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2-x^5\,{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2+64\,x^4-20\,x^5+x^6}{20\,\left (16\,x^2-x^3\right )}+\frac {\ln \relax (x)\,\left (16\,x^4\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-x^5\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )\right )}{10\,\left (16\,x^2-x^3\right )}}{4\,\ln \relax (x)\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-x+4\,{\ln \relax (x)}^2+{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 26, normalized size = 0.79 \begin {gather*} \frac {x \log {\relax (x )}}{5 \log {\left (\frac {- \frac {x}{5} + \frac {4 \log {\relax (x )}}{5}}{2 x \log {\relax (x )}} \right )}} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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