Optimal. Leaf size=32 \[ 2 e^{-\left (-3+\frac {\left (9-\log \left (\frac {5}{x}\right )\right ) (x+\log (x))}{x}\right )^2} x+\log (2) \]
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Rubi [F]
time = 63.18, 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 {\exp \left (-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}\right ) \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int 488281250 \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} \left (-x (108+11 x)+6 (-27+13 x) \log (x)+144 \log ^2(x)+2 \log ^2\left (\frac {5}{x}\right ) (-1+\log (x)) (x+\log (x))+2 \log \left (\frac {5}{x}\right ) \left (x (15+x)+(18-13 x) \log (x)-17 \log ^2(x)\right )\right ) \, dx\\ &=488281250 \int \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} \left (-x (108+11 x)+6 (-27+13 x) \log (x)+144 \log ^2(x)+2 \log ^2\left (\frac {5}{x}\right ) (-1+\log (x)) (x+\log (x))+2 \log \left (\frac {5}{x}\right ) \left (x (15+x)+(18-13 x) \log (x)-17 \log ^2(x)\right )\right ) \, dx\\ &=488281250 \int \left (-\exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{1-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} (108+11 x)+6 \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} (-27+13 x) \log (x)+144 \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} \log ^2(x)+2 \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} \log ^2\left (\frac {5}{x}\right ) (-1+\log (x)) (x+\log (x))+2 \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} \log \left (\frac {5}{x}\right ) \left (15 x+x^2+18 \log (x)-13 x \log (x)-17 \log ^2(x)\right )\right ) \, dx\\ &=-\left (488281250 \int \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{1-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} (108+11 x) \, dx\right )+976562500 \int \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} \log ^2\left (\frac {5}{x}\right ) (-1+\log (x)) (x+\log (x)) \, dx+976562500 \int \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} \log \left (\frac {5}{x}\right ) \left (15 x+x^2+18 \log (x)-13 x \log (x)-17 \log ^2(x)\right ) \, dx+2929687500 \int \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} (-27+13 x) \log (x) \, dx+70312500000 \int \exp \left (-36-\log ^2\left (\frac {5}{x}\right )-\frac {\left (-9+\log \left (\frac {5}{x}\right )\right )^2 \log ^2(x)}{x^2}\right ) x^{-\frac {2 \left (54+7 x-15 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right )}{x}} \log ^2(x) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(32)=64\).
time = 0.37, size = 160, normalized size = 5.00 \begin {gather*} 2\ 5^{12+\frac {6 \left (5 x \log (x)+3 \left (\log \left (\frac {5}{x}\right )+\log (x)\right ) \left (9+\log \left (\frac {5}{x}\right )+\log (x)\right )\right )}{x^2}} e^{-\frac {18 \log ^3\left (\frac {5}{x}\right )+162 \log \left (\frac {5}{x}\right ) \log (x)+\log ^2\left (\frac {5}{x}\right ) \left (162+x^2+36 \log (x)+\log ^2(x)\right )+9 \left (4 x^2+9 \log ^2(x)\right )}{x^2}} \left (\frac {1}{x}\right )^{\frac {6 \left (5 x \log (x)+3 \left (\log \left (\frac {5}{x}\right )+\log (x)\right ) \left (9+\log \left (\frac {5}{x}\right )+\log (x)\right )\right )}{x^2}} x^{-\frac {108+11 x+2 \log ^2\left (\frac {5}{x}\right )}{x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 14.56, size = 40, normalized size = 1.25
method | result | size |
risch | \(2 x \,{\mathrm e}^{-\frac {\left (-\ln \left (x \right )^{2}+\ln \left (5\right ) \ln \left (x \right )-x \ln \left (x \right )+x \ln \left (5\right )-9 \ln \left (x \right )-6 x \right )^{2}}{x^{2}}}\) | \(40\) |
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. 144 vs.
\(2 (29) = 58\).
time = 0.73, size = 144, normalized size = 4.50 \begin {gather*} \frac {488281250 \, e^{\left (-\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) - \frac {2 \, \log \left (5\right )^{2} \log \left (x\right )}{x} + \frac {4 \, \log \left (5\right ) \log \left (x\right )^{2}}{x} - \frac {\log \left (5\right )^{2} \log \left (x\right )^{2}}{x^{2}} - \log \left (x\right )^{2} - \frac {2 \, \log \left (x\right )^{3}}{x} + \frac {2 \, \log \left (5\right ) \log \left (x\right )^{3}}{x^{2}} - \frac {\log \left (x\right )^{4}}{x^{2}} + \frac {30 \, \log \left (5\right ) \log \left (x\right )}{x} - \frac {30 \, \log \left (x\right )^{2}}{x} + \frac {18 \, \log \left (5\right ) \log \left (x\right )^{2}}{x^{2}} - \frac {18 \, \log \left (x\right )^{3}}{x^{2}} - \frac {108 \, \log \left (x\right )}{x} - \frac {81 \, \log \left (x\right )^{2}}{x^{2}} - 36\right )}}{x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs.
\(2 (29) = 58\).
time = 12.46, size = 110, normalized size = 3.44 \begin {gather*} 2 \, x e^{\left (\frac {2 \, {\left (x + \log \left (5\right ) + 9\right )} \log \left (\frac {5}{x}\right )^{3} - \log \left (\frac {5}{x}\right )^{4} - {\left (x^{2} + 2 \, {\left (x + 18\right )} \log \left (5\right ) + \log \left (5\right )^{2} + 30 \, x + 81\right )} \log \left (\frac {5}{x}\right )^{2} - 36 \, x^{2} - 108 \, x \log \left (5\right ) - 81 \, \log \left (5\right )^{2} + 6 \, {\left (2 \, x^{2} + {\left (5 \, x + 27\right )} \log \left (5\right ) + 3 \, \log \left (5\right )^{2} + 18 \, x\right )} \log \left (\frac {5}{x}\right )}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (24) = 48\).
time = 0.78, size = 90, normalized size = 2.81 \begin {gather*} 2 x e^{- \frac {x^{2} \left (- \log {\left (x \right )} + \log {\left (5 \right )}\right )^{2} - 12 x^{2} \left (- \log {\left (x \right )} + \log {\left (5 \right )}\right ) + 36 x^{2} + \left (2 x \left (- \log {\left (x \right )} + \log {\left (5 \right )}\right )^{2} - 30 x \left (- \log {\left (x \right )} + \log {\left (5 \right )}\right ) + 108 x\right ) \log {\left (x \right )} + \left (\left (- \log {\left (x \right )} + \log {\left (5 \right )}\right )^{2} + 18 \log {\left (x \right )} - 18 \log {\left (5 \right )} + 81\right ) \log {\left (x \right )}^{2}}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs.
\(2 (29) = 58\).
time = 0.77, size = 139, normalized size = 4.34 \begin {gather*} 2 \, x e^{\left (-\frac {x^{2} \log \left (5\right )^{2} - 2 \, x^{2} \log \left (5\right ) \log \left (x\right ) + 2 \, x \log \left (5\right )^{2} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 4 \, x \log \left (5\right ) \log \left (x\right )^{2} + \log \left (5\right )^{2} \log \left (x\right )^{2} + 2 \, x \log \left (x\right )^{3} - 2 \, \log \left (5\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4} - 12 \, x^{2} \log \left (5\right ) + 12 \, x^{2} \log \left (x\right ) - 30 \, x \log \left (5\right ) \log \left (x\right ) + 30 \, x \log \left (x\right )^{2} - 18 \, \log \left (5\right ) \log \left (x\right )^{2} + 18 \, \log \left (x\right )^{3} + 36 \, x^{2} + 108 \, x \log \left (x\right ) + 81 \, \log \left (x\right )^{2}}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.26, size = 186, normalized size = 5.81 \begin {gather*} \frac {488281250\,5^{\frac {18\,{\ln \left (x\right )}^2}{x^2}}\,x^{\frac {30\,\ln \left (\frac {1}{x}\right )}{x}}\,x^{\frac {30\,\ln \left (5\right )}{x}}\,{\mathrm {e}}^{-36}\,{\mathrm {e}}^{-{\ln \left (\frac {1}{x}\right )}^2}\,{\mathrm {e}}^{-\frac {{\ln \left (\frac {1}{x}\right )}^2\,{\ln \left (x\right )}^2}{x^2}}\,{\mathrm {e}}^{-{\ln \left (5\right )}^2}\,{\mathrm {e}}^{-\frac {{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2}{x^2}}\,{\mathrm {e}}^{-\frac {81\,{\ln \left (x\right )}^2}{x^2}}\,{\left (\frac {1}{x}\right )}^{\frac {18\,{\ln \left (x\right )}^2}{x^2}}}{x^{108/x}\,x^{\frac {2\,{\ln \left (\frac {1}{x}\right )}^2}{x}}\,x^{\frac {4\,\ln \left (\frac {1}{x}\right )\,\ln \left (5\right )}{x}}\,x^{11}\,x^{\frac {2\,{\ln \left (5\right )}^2}{x}}\,{\left (\frac {1}{x}\right )}^{2\,\ln \left (5\right )}\,{\left (\frac {1}{x}\right )}^{\frac {2\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x^2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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