Optimal. Leaf size=25 \[ e^{1-\log ^2(-4 x) \log (x)} \left (x-e^4 x\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(25)=50\).
time = 0.26, antiderivative size = 72, normalized size of antiderivative = 2.88, number of steps
used = 3, number of rules used = 2, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6, 2326}
\begin {gather*} \frac {e x^{-\log ^2(-4 x)} \left (\left (1-e^4\right )^2 x \log ^2(-4 x)+2 \left (1-e^4\right )^2 x \log (x) \log (-4 x)\right )}{\frac {\log ^2(-4 x)}{x}+\frac {2 \log (x) \log (-4 x)}{x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int e^{1-\log ^2(-4 x) \log (x)} \left (2 e^8 x+\left (2-4 e^4\right ) x+\left (-x+2 e^4 x-e^8 x\right ) \log ^2(-4 x)+\left (-2 x+4 e^4 x-2 e^8 x\right ) \log (-4 x) \log (x)\right ) \, dx\\ &=\int e^{1-\log ^2(-4 x) \log (x)} \left (\left (2-4 e^4+2 e^8\right ) x+\left (-x+2 e^4 x-e^8 x\right ) \log ^2(-4 x)+\left (-2 x+4 e^4 x-2 e^8 x\right ) \log (-4 x) \log (x)\right ) \, dx\\ &=\frac {e x^{-\log ^2(-4 x)} \left (\left (1-e^4\right )^2 x \log ^2(-4 x)+2 \left (1-e^4\right )^2 x \log (-4 x) \log (x)\right )}{\frac {\log ^2(-4 x)}{x}+\frac {2 \log (-4 x) \log (x)}{x}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 21, normalized size = 0.84 \begin {gather*} e \left (-1+e^4\right )^2 x^{2-\log ^2(-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 = 1.45, size = 37, normalized size = 1.48
method | result | size |
risch | \(x^{2} \left ({\mathrm e}^{8}-2 \,{\mathrm e}^{4}+1\right ) x^{-\left (i \pi \,\mathrm {csgn}\left (i x \right )+\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}} {\mathrm e}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.56, size = 57, normalized size = 2.28 \begin {gather*} x^{2} {\left (e^{9} - 2 \, e^{5} + e\right )} e^{\left (\pi ^{2} \log \left (x\right ) - 4 i \, \pi \log \left (2\right ) \log \left (x\right ) - 4 \, \log \left (2\right )^{2} \log \left (x\right ) - 2 i \, \pi \log \left (x\right )^{2} - 4 \, \log \left (2\right ) \log \left (x\right )^{2} - \log \left (x\right )^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 48, normalized size = 1.92 \begin {gather*} {\left (x^{2} e^{8} - 2 \, x^{2} e^{4} + x^{2}\right )} \cos \left (\pi \log \left (-4 \, x\right )^{2}\right ) e^{\left (2 \, \log \left (2\right ) \log \left (-4 \, x\right )^{2} - \log \left (-4 \, x\right )^{3} + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.44, size = 36, normalized size = 1.44 \begin {gather*} \left (- 2 x^{2} e^{4} + x^{2} + x^{2} e^{8}\right ) e^{- \left (\log {\left (x \right )} + \log {\left (4 \right )} + i \pi \right )^{2} \log {\left (x \right )} + 1} \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. 675 vs.
\(2 (24) = 48\).
time = 0.46, size = 675, normalized size = 27.00 \begin {gather*} -\frac {x^{2} e^{\left (-6 \, \pi ^{2} \log \left (2\right ) \mathrm {sgn}\left (x\right ) - \frac {11}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) + 6 \, \pi ^{2} \log \left (2\right ) + \frac {19}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right )^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right ) \log \left ({\left | x \right |}\right )^{2} - \log \left ({\left | x \right |}\right )^{3} + 9\right )} \tan \left (\frac {9}{4} \, \pi ^{3} \mathrm {sgn}\left (x\right ) - \pi \log \left (2\right )^{2} \mathrm {sgn}\left (x\right ) - 2 \, \pi \log \left (2\right ) \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {3}{4} \, \pi \log \left ({\left | x \right |}\right )^{2} \mathrm {sgn}\left (x\right ) - \frac {9}{4} \, \pi ^{3} + \pi \log \left (2\right )^{2} + 6 \, \pi \log \left (2\right ) \log \left ({\left | x \right |}\right ) + \frac {11}{4} \, \pi \log \left ({\left | x \right |}\right )^{2}\right )^{2} - 2 \, x^{2} e^{\left (-6 \, \pi ^{2} \log \left (2\right ) \mathrm {sgn}\left (x\right ) - \frac {11}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) + 6 \, \pi ^{2} \log \left (2\right ) + \frac {19}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right )^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right ) \log \left ({\left | x \right |}\right )^{2} - \log \left ({\left | x \right |}\right )^{3} + 5\right )} \tan \left (\frac {9}{4} \, \pi ^{3} \mathrm {sgn}\left (x\right ) - \pi \log \left (2\right )^{2} \mathrm {sgn}\left (x\right ) - 2 \, \pi \log \left (2\right ) \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {3}{4} \, \pi \log \left ({\left | x \right |}\right )^{2} \mathrm {sgn}\left (x\right ) - \frac {9}{4} \, \pi ^{3} + \pi \log \left (2\right )^{2} + 6 \, \pi \log \left (2\right ) \log \left ({\left | x \right |}\right ) + \frac {11}{4} \, \pi \log \left ({\left | x \right |}\right )^{2}\right )^{2} + x^{2} e^{\left (-6 \, \pi ^{2} \log \left (2\right ) \mathrm {sgn}\left (x\right ) - \frac {11}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) + 6 \, \pi ^{2} \log \left (2\right ) + \frac {19}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right )^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right ) \log \left ({\left | x \right |}\right )^{2} - \log \left ({\left | x \right |}\right )^{3} + 1\right )} \tan \left (\frac {9}{4} \, \pi ^{3} \mathrm {sgn}\left (x\right ) - \pi \log \left (2\right )^{2} \mathrm {sgn}\left (x\right ) - 2 \, \pi \log \left (2\right ) \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {3}{4} \, \pi \log \left ({\left | x \right |}\right )^{2} \mathrm {sgn}\left (x\right ) - \frac {9}{4} \, \pi ^{3} + \pi \log \left (2\right )^{2} + 6 \, \pi \log \left (2\right ) \log \left ({\left | x \right |}\right ) + \frac {11}{4} \, \pi \log \left ({\left | x \right |}\right )^{2}\right )^{2} - x^{2} e^{\left (-6 \, \pi ^{2} \log \left (2\right ) \mathrm {sgn}\left (x\right ) - \frac {11}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) + 6 \, \pi ^{2} \log \left (2\right ) + \frac {19}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right )^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right ) \log \left ({\left | x \right |}\right )^{2} - \log \left ({\left | x \right |}\right )^{3} + 9\right )} + 2 \, x^{2} e^{\left (-6 \, \pi ^{2} \log \left (2\right ) \mathrm {sgn}\left (x\right ) - \frac {11}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) + 6 \, \pi ^{2} \log \left (2\right ) + \frac {19}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right )^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right ) \log \left ({\left | x \right |}\right )^{2} - \log \left ({\left | x \right |}\right )^{3} + 5\right )} - x^{2} e^{\left (-6 \, \pi ^{2} \log \left (2\right ) \mathrm {sgn}\left (x\right ) - \frac {11}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) + 6 \, \pi ^{2} \log \left (2\right ) + \frac {19}{2} \, \pi ^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right )^{2} \log \left ({\left | x \right |}\right ) - 4 \, \log \left (2\right ) \log \left ({\left | x \right |}\right )^{2} - \log \left ({\left | x \right |}\right )^{3} + 1\right )}}{\tan \left (\frac {9}{4} \, \pi ^{3} \mathrm {sgn}\left (x\right ) - \pi \log \left (2\right )^{2} \mathrm {sgn}\left (x\right ) - 2 \, \pi \log \left (2\right ) \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {3}{4} \, \pi \log \left ({\left | x \right |}\right )^{2} \mathrm {sgn}\left (x\right ) - \frac {9}{4} \, \pi ^{3} + \pi \log \left (2\right )^{2} + 6 \, \pi \log \left (2\right ) \log \left ({\left | x \right |}\right ) + \frac {11}{4} \, \pi \log \left ({\left | x \right |}\right )^{2}\right )^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.53, size = 21, normalized size = 0.84 \begin {gather*} x^{2-{\ln \left (-4\,x\right )}^2}\,\mathrm {e}\,{\left ({\mathrm {e}}^4-1\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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