Optimal. Leaf size=22 \[ e^{\frac {\log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^2}} \]
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Rubi [F]
time = 6.40, 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 {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{\left (20 x^5-x^6\right ) \log (-20+x)+\left (-40 x^3+2 x^4\right ) \log ^2(-20+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 {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{(20-x) x^3 \left (x^2-2 \log (-20+x)\right ) \log (-20+x)} \, dx\\ &=\int \left (\frac {2 \exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) (-x-40 \log (-20+x)+2 x \log (-20+x)) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) x \left (x^2-2 \log (-20+x)\right ) \log (-20+x)}-\frac {2 \exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^3}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) (-x-40 \log (-20+x)+2 x \log (-20+x)) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) x \left (x^2-2 \log (-20+x)\right ) \log (-20+x)} \, dx-2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^3} \, dx\\ &=-\left (2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^3} \, dx\right )+2 \int \left (\frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) (-x-40 \log (-20+x)+2 x \log (-20+x)) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{20 (-20+x) \left (x^2-2 \log (-20+x)\right ) \log (-20+x)}-\frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) (-x-40 \log (-20+x)+2 x \log (-20+x)) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{20 x \left (x^2-2 \log (-20+x)\right ) \log (-20+x)}\right ) \, dx\\ &=\frac {1}{10} \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) (-x-40 \log (-20+x)+2 x \log (-20+x)) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right ) \log (-20+x)} \, dx-\frac {1}{10} \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) (-x-40 \log (-20+x)+2 x \log (-20+x)) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x \left (x^2-2 \log (-20+x)\right ) \log (-20+x)} \, dx-2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^3} \, dx\\ &=-\left (\frac {1}{10} \int \left (\frac {2 \exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x^2-2 \log (-20+x)}-\frac {40 \exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x \left (x^2-2 \log (-20+x)\right )}-\frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{\left (x^2-2 \log (-20+x)\right ) \log (-20+x)}\right ) \, dx\right )+\frac {1}{10} \int \left (-\frac {40 \exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right )}+\frac {2 \exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) x \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right )}-\frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) x \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right ) \log (-20+x)}\right ) \, dx-2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^3} \, dx\\ &=\frac {1}{10} \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{\left (x^2-2 \log (-20+x)\right ) \log (-20+x)} \, dx-\frac {1}{10} \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) x \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right ) \log (-20+x)} \, dx-\frac {1}{5} \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x^2-2 \log (-20+x)} \, dx+\frac {1}{5} \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) x \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right )} \, dx-2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^3} \, dx-4 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right )} \, dx+4 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x \left (x^2-2 \log (-20+x)\right )} \, dx\\ &=\frac {1}{10} \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{\left (x^2-2 \log (-20+x)\right ) \log (-20+x)} \, dx-\frac {1}{10} \int \left (\frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{\left (x^2-2 \log (-20+x)\right ) \log (-20+x)}+\frac {20 \exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right ) \log (-20+x)}\right ) \, dx-\frac {1}{5} \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x^2-2 \log (-20+x)} \, dx+\frac {1}{5} \int \left (\frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x^2-2 \log (-20+x)}+\frac {20 \exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right )}\right ) \, dx-2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^3} \, dx-4 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right )} \, dx+4 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x \left (x^2-2 \log (-20+x)\right )} \, dx\\ &=-\left (2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{(-20+x) \left (x^2-2 \log (-20+x)\right ) \log (-20+x)} \, dx\right )-2 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^3} \, dx+4 \int \frac {\exp \left (\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}\right ) \log \left (2-\frac {x^2}{\log (-20+x)}\right )}{x \left (x^2-2 \log (-20+x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 22, normalized size = 1.00 \begin {gather*} e^{\frac {\log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^2}} \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 = 0.42, size = 210, normalized size = 9.55
method | result | size |
risch | \({\mathrm e}^{\frac {\left (-i \pi \mathrm {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (x -20\right )}\right )-i \pi \mathrm {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right )^{2} \mathrm {csgn}\left (i \left (-2 \ln \left (x -20\right )+x^{2}\right )\right )+i \pi \,\mathrm {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x -20\right )}\right ) \mathrm {csgn}\left (i \left (-2 \ln \left (x -20\right )+x^{2}\right )\right )+2 i \pi \mathrm {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right )^{2}-2 i \pi +2 \ln \left (\ln \left (x -20\right )\right )-2 \ln \left (-2 \ln \left (x -20\right )+x^{2}\right )\right )^{2}}{4 x^{2}}}\) | \(210\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (21) = 42\).
time = 0.43, size = 55, normalized size = 2.50 \begin {gather*} e^{\left (\frac {\log \left (-x^{2} + 2 \, \log \left (x - 20\right )\right )^{2}}{x^{2}} - \frac {2 \, \log \left (-x^{2} + 2 \, \log \left (x - 20\right )\right ) \log \left (\log \left (x - 20\right )\right )}{x^{2}} + \frac {\log \left (\log \left (x - 20\right )\right )^{2}}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 26, normalized size = 1.18 \begin {gather*} e^{\left (\frac {\log \left (-\frac {x^{2} - 2 \, \log \left (x - 20\right )}{\log \left (x - 20\right )}\right )^{2}}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.85, size = 22, normalized size = 1.00 \begin {gather*} e^{\frac {\log {\left (\frac {- x^{2} + 2 \log {\left (x - 20 \right )}}{\log {\left (x - 20 \right )}} \right )}^{2}}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.67, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (\frac {\log \left (-\frac {x^{2}}{\log \left (x - 20\right )} + 2\right )^{2}}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.46, size = 27, normalized size = 1.23 \begin {gather*} {\mathrm {e}}^{\frac {{\ln \left (\frac {2\,\ln \left (x-20\right )-x^2}{\ln \left (x-20\right )}\right )}^2}{x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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