Optimal. Leaf size=26 \[ \log ^2\left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right ) \]
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Rubi [F] time = 2.29, 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 (-25 x+9 x^2-8 x^3+16 x^4-50 \log (x)\right ) \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{4 x^4-4 x^5+8 x^3 \log (x)} \, 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 (-25 x+9 x^2-8 x^3+16 x^4-50 \log (x)\right ) \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{4 x^3 \left (x-x^2+2 \log (x)\right )} \, dx\\ &=\frac {1}{4} \int \frac {\left (-25 x+9 x^2-8 x^3+16 x^4-50 \log (x)\right ) \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{x^3 \left (x-x^2+2 \log (x)\right )} \, dx\\ &=\frac {1}{4} \int \left (\frac {8 \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{-x+x^2-2 \log (x)}+\frac {25 \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{x^2 \left (-x+x^2-2 \log (x)\right )}-\frac {9 \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{x \left (-x+x^2-2 \log (x)\right )}-\frac {16 x \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{-x+x^2-2 \log (x)}+\frac {50 \log (x) \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{x^3 \left (-x+x^2-2 \log (x)\right )}\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{-x+x^2-2 \log (x)} \, dx-\frac {9}{4} \int \frac {\log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{x \left (-x+x^2-2 \log (x)\right )} \, dx-4 \int \frac {x \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{-x+x^2-2 \log (x)} \, dx+\frac {25}{4} \int \frac {\log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{x^2 \left (-x+x^2-2 \log (x)\right )} \, dx+\frac {25}{2} \int \frac {\log (x) \log \left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right )}{x^3 \left (-x+x^2-2 \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.26, size = 26, normalized size = 1.00 \begin {gather*} \log ^2\left (\frac {e^{\frac {25}{16 x^2}}}{x-x^2+2 \log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 24, normalized size = 0.92 \begin {gather*} \log \left (-\frac {e^{\left (\frac {25}{16 \, x^{2}}\right )}}{x^{2} - x - 2 \, \log \relax (x)}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.28, size = 60, normalized size = 2.31 \begin {gather*} \log \left (x^{2} - x - 2 \, \log \relax (x)\right )^{2} - 2 i \, \pi \log \left (-x^{2} + x + 2 \, \log \relax (x)\right ) - \frac {25 \, \log \left (x^{2} - x - 2 \, \log \relax (x)\right )}{8 \, x^{2}} + \frac {25 \, {\left (32 i \, \pi x^{2} + 25\right )}}{256 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 569, normalized size = 21.88
method | result | size |
risch | \(-\frac {\left (16 \ln \left (-2 \ln \relax (x )+x^{2}-x \right ) x^{2}-25\right ) \ln \left ({\mathrm e}^{\frac {25}{16 x^{2}}}\right )}{8 x^{2}}+\frac {-512 i \pi \ln \left (-\frac {x^{2}}{2}+\frac {x}{2}+\ln \relax (x )\right ) x^{4}-400 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{2 \ln \relax (x )-x^{2}+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {25}{16 x^{2}}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )+256 i \pi \ln \left (-\frac {x^{2}}{2}+\frac {x}{2}+\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )^{3} x^{4}-400 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )^{3}-800 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )^{2}-256 i \pi \ln \left (-\frac {x^{2}}{2}+\frac {x}{2}+\ln \relax (x )\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {25}{16 x^{2}}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )^{2} x^{4}+256 i \pi \ln \left (-\frac {x^{2}}{2}+\frac {x}{2}+\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (x )-x^{2}+x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {25}{16 x^{2}}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right ) x^{4}+800 i \pi \,x^{2}-400 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{2 \ln \relax (x )-x^{2}+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )^{2}+256 i \pi \ln \left (-\frac {x^{2}}{2}+\frac {x}{2}+\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (x )-x^{2}+x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )^{2} x^{4}+400 i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{\frac {25}{16 x^{2}}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )^{2}+512 i \pi \ln \left (-\frac {x^{2}}{2}+\frac {x}{2}+\ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {25}{16 x^{2}}}}{2 \ln \relax (x )-x^{2}+x}\right )^{2} x^{4}-625+256 \ln \left (-2 \ln \relax (x )+x^{2}-x \right )^{2} x^{4}}{256 x^{4}}\) | \(569\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 105, normalized size = 4.04 \begin {gather*} \frac {1}{8} \, {\left (\frac {25}{x^{2}} - 16 \, \log \left (-\frac {1}{2} \, x^{2} + \frac {1}{2} \, x + \log \relax (x)\right )\right )} \log \left (-\frac {e^{\left (\frac {25}{16 \, x^{2}}\right )}}{x^{2} - x - 2 \, \log \relax (x)}\right ) - \frac {256 \, x^{4} \log \left (-x^{2} + x + 2 \, \log \relax (x)\right )^{2} + 800 \, x^{2} \log \relax (2) - 32 \, {\left (16 \, x^{4} \log \relax (2) + 25 \, x^{2}\right )} \log \left (-x^{2} + x + 2 \, \log \relax (x)\right ) + 625}{256 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.84, size = 28, normalized size = 1.08 \begin {gather*} \frac {{\left (16\,x^2\,\ln \left (\frac {1}{x+2\,\ln \relax (x)-x^2}\right )+25\right )}^2}{256\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 20, normalized size = 0.77 \begin {gather*} \log {\left (\frac {e^{\frac {25}{16 x^{2}}}}{- x^{2} + x + 2 \log {\relax (x )}} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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