Optimal. Leaf size=17 \[ e^{-(2+\log (x (9+\log (3)) \log (x)))^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(17)=34\).
time = 0.16, antiderivative size = 48, normalized size of antiderivative = 2.82, number of steps
used = 3, number of rules used = 3, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6, 12, 2326}
\begin {gather*} \frac {e^{-\log ^2(x (9+\log (3)) \log (x))-4} (\log (x)+1)}{x^4 (9+\log (3))^3 \log ^4(x) ((9+\log (3)) \log (x)+9+\log (3))} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x^5 (9+\log (3))^4 \log ^5(x)} \, dx\\ &=\frac {\int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x^5 \log ^5(x)} \, dx}{(9+\log (3))^4}\\ &=\frac {e^{-4-\log ^2(x (9+\log (3)) \log (x))} (1+\log (x))}{x^4 (9+\log (3))^3 \log ^4(x) (9+\log (3)+(9+\log (3)) \log (x))}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 31, normalized size = 1.82 \begin {gather*} \frac {e^{-4-\log ^2(x (9+\log (3)) \log (x))}}{x^4 (9+\log (3))^4 \log ^4(x)} \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 = 0.69, size = 567, normalized size = 33.35
method | result | size |
risch | \(\frac {\ln \left (x \right )^{-2 \ln \left (9+\ln \left (3\right )\right )} \left (9+\ln \left (3\right )\right )^{i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (x \right )\right )} \left (9+\ln \left (3\right )\right )^{-i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right )} x^{-i \pi \,\mathrm {csgn}\left (i x \right )} \left (9+\ln \left (3\right )\right )^{-i \pi \,\mathrm {csgn}\left (i x \right )} \left (9+\ln \left (3\right )\right )^{i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right )} \ln \left (x \right )^{-i \pi \,\mathrm {csgn}\left (i x \right )} \ln \left (x \right )^{-i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right )} x^{i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right )} \ln \left (x \right )^{i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (x \right )\right )} x^{-2 \ln \left (9+\ln \left (3\right )\right )} \ln \left (x \right )^{-2 \ln \left (x \right )} x^{-i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right )} \ln \left (x \right )^{i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right )} x^{i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (x \right )\right )} {\mathrm e}^{-4-2 i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2}+2 i \pi \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{3}-\ln \left (\ln \left (x \right )\right )^{2}-\ln \left (x \right )^{2}+2 i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (x \right )\right )-\frac {\pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{5}}{2}+\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{4}}{4}-\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{5}}{2}+\frac {\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{4}}{4}+\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{6}}{4}-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2}-\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{3}}{2}+\pi ^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{4}+\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2}}{4}-\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{3}}{2}-\ln \left (9+\ln \left (3\right )\right )^{2}}}{\ln \left (x \right )^{4} x^{4} \left (9+\ln \left (3\right )\right )^{4}}\) | \(567\) |
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. 83 vs.
\(2 (16) = 32\).
time = 0.58, size = 83, normalized size = 4.88 \begin {gather*} \frac {e^{\left (-\log \left (x\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (3\right ) + 9\right ) - \log \left (\log \left (3\right ) + 9\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 2 \, \log \left (\log \left (3\right ) + 9\right ) \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} - 4\right )}}{{\left (\log \left (3\right )^{4} + 36 \, \log \left (3\right )^{3} + 486 \, \log \left (3\right )^{2} + 2916 \, \log \left (3\right ) + 6561\right )} x^{4} \log \left (x\right )^{4}} \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. 33 vs.
\(2 (16) = 32\).
time = 0.39, size = 33, normalized size = 1.94 \begin {gather*} e^{\left (-\log \left ({\left (x \log \left (3\right ) + 9 \, x\right )} \log \left (x\right )\right )^{2} - 4 \, \log \left ({\left (x \log \left (3\right ) + 9 \, x\right )} \log \left (x\right )\right ) - 4\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. 87 vs.
\(2 (17) = 34\).
time = 0.24, size = 87, normalized size = 5.12 \begin {gather*} \frac {e^{- \log {\left (\left (x \log {\left (3 \right )} + 9 x\right ) \log {\left (x \right )} \right )}^{2} - 4}}{x^{4} \log {\left (3 \right )}^{4} \log {\left (x \right )}^{4} + 36 x^{4} \log {\left (3 \right )}^{3} \log {\left (x \right )}^{4} + 486 x^{4} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{4} + 2916 x^{4} \log {\left (3 \right )} \log {\left (x \right )}^{4} + 6561 x^{4} \log {\left (x \right )}^{4}} \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. 35 vs.
\(2 (16) = 32\).
time = 0.46, size = 35, normalized size = 2.06 \begin {gather*} e^{\left (-\log \left (x \log \left (3\right ) \log \left (x\right ) + 9 \, x \log \left (x\right )\right )^{2} - 4 \, \log \left (x \log \left (3\right ) \log \left (x\right ) + 9 \, x \log \left (x\right )\right ) - 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.81, size = 82, normalized size = 4.82 \begin {gather*} \frac {{\mathrm {e}}^{-4}\,{\mathrm {e}}^{-{\ln \left (9\,x\,\ln \left (x\right )+x\,\ln \left (3\right )\,\ln \left (x\right )\right )}^2}}{6561\,x^4\,{\ln \left (x\right )}^4+486\,x^4\,{\ln \left (3\right )}^2\,{\ln \left (x\right )}^4+36\,x^4\,{\ln \left (3\right )}^3\,{\ln \left (x\right )}^4+x^4\,{\ln \left (3\right )}^4\,{\ln \left (x\right )}^4+2916\,x^4\,\ln \left (3\right )\,{\ln \left (x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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