Optimal. Leaf size=25 \[ e^{\left (-\frac {1}{4}-\log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(25)=50\).
time = 1.00, antiderivative size = 107, normalized size of antiderivative = 4.28, number of steps
used = 3, number of rules used = 3, integrand size = 118, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2306, 12,
2326} \begin {gather*} \frac {\sqrt {2} \left (e^x+e^5\right ) \left (e^x (1-x)+e^5\right ) e^{\frac {1}{16} \left (16 \log ^2\left (2 \log \left (\frac {e^x+e^5}{x}\right )\right )+1\right )} \sqrt {\log \left (\frac {e^x+e^5}{x}\right )}}{\left (\frac {e^x+e^5}{x^2}-\frac {e^x}{x}\right ) x \left (e^x x+e^5 x\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2306
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\sqrt {2} e^{\frac {1}{16} \left (1+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \sqrt {\log \left (\frac {e^5+e^x}{x}\right )}} \, dx\\ &=\sqrt {2} \int \frac {e^{\frac {1}{16} \left (1+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \sqrt {\log \left (\frac {e^5+e^x}{x}\right )}} \, dx\\ &=\frac {\sqrt {2} e^{\frac {1}{16} \left (1+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (e^5+e^x\right ) \left (e^5+e^x (1-x)\right ) \sqrt {\log \left (\frac {e^5+e^x}{x}\right )}}{\left (\frac {e^5+e^x}{x^2}-\frac {e^x}{x}\right ) x \left (e^5 x+e^x x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 45, normalized size = 1.80 \begin {gather*} \sqrt {2} e^{\frac {1}{16}+\log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )} \sqrt {\log \left (\frac {e^5+e^x}{x}\right )} \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 = 1.08, size = 160, normalized size = 6.40
method | result | size |
risch | \(\sqrt {-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right ) \left (-\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right )+\mathrm {csgn}\left (i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )\right )\right )}\, {\mathrm e}^{\ln \left (-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right ) \left (-\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right )+\mathrm {csgn}\left (i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )\right )\right )\right )^{2}+\frac {1}{16}}\) | \(160\) |
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. 54 vs.
\(2 (22) = 44\).
time = 0.69, size = 54, normalized size = 2.16 \begin {gather*} \sqrt {2} \sqrt {-\log \left (x\right ) + \log \left (e^{5} + e^{x}\right )} e^{\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (-\log \left (x\right ) + \log \left (e^{5} + e^{x}\right )\right ) + \log \left (-\log \left (x\right ) + \log \left (e^{5} + e^{x}\right )\right )^{2} + \frac {1}{16}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 33, normalized size = 1.32 \begin {gather*} e^{\left (\log \left (2 \, \log \left (\frac {e^{5} + e^{x}}{x}\right )\right )^{2} + \frac {1}{2} \, \log \left (2 \, \log \left (\frac {e^{5} + e^{x}}{x}\right )\right ) + \frac {1}{16}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.08, size = 39, normalized size = 1.56 \begin {gather*} {\mathrm {e}}^{1/16}\,{\mathrm {e}}^{{\ln \left (2\,\ln \left (\frac {1}{x}\right )+\ln \left ({\left ({\mathrm {e}}^5+{\mathrm {e}}^x\right )}^2\right )\right )}^2}\,\sqrt {2\,\ln \left (\frac {1}{x}\right )+\ln \left ({\left ({\mathrm {e}}^5+{\mathrm {e}}^x\right )}^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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