Optimal. Leaf size=24 \[ 4+e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x}+x \]
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Rubi [F]
time = 4.43, 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 {e^x x+5 x \log \left (x^2\right )+e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x} \left (e^x x+5 x \log \left (x^2\right )+e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )} \left (-40-4 e^x x\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )\right )}{e^x x+5 x \log \left (x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}} \left (e^{e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}}+e^x\right )-\frac {4 \exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \left (10+e^x x\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )}{x \left (e^x+5 \log \left (x^2\right )\right )}\right ) \, dx\\ &=-\left (4 \int \frac {\exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \left (10+e^x x\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )}{x \left (e^x+5 \log \left (x^2\right )\right )} \, dx\right )+\int e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}} \left (e^{e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}}+e^x\right ) \, dx\\ &=-\left (4 \int \left (\exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )-\frac {5 \exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \left (-2+x \log \left (x^2\right )\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )}{x \left (e^x+5 \log \left (x^2\right )\right )}\right ) \, dx\right )+\int \left (1+e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x}\right ) \, dx\\ &=x-4 \int \exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right ) \, dx+20 \int \frac {\exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \left (-2+x \log \left (x^2\right )\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )}{x \left (e^x+5 \log \left (x^2\right )\right )} \, dx+\int e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x} \, dx\\ &=x-4 \int \exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right ) \, dx+20 \int \left (-\frac {2 \exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )}{x \left (e^x+5 \log \left (x^2\right )\right )}+\frac {\exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \log \left (x^2\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )}{e^x+5 \log \left (x^2\right )}\right ) \, dx+\int e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x} \, dx\\ &=x-4 \int \exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right ) \, dx+20 \int \frac {\exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \log \left (x^2\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )}{e^x+5 \log \left (x^2\right )} \, dx-40 \int \frac {\exp \left (-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x+\log ^4\left (e^x+5 \log \left (x^2\right )\right )\right ) \log ^3\left (e^x+5 \log \left (x^2\right )\right )}{x \left (e^x+5 \log \left (x^2\right )\right )} \, dx+\int e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 23, normalized size = 0.96 \begin {gather*} e^{-e^{\log ^4\left (e^x+5 \log \left (x^2\right )\right )}+x}+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 = 0.32, size = 48, normalized size = 2.00
method | result | size |
risch | \(x +{\mathrm e}^{x} {\mathrm e}^{-{\mathrm e}^{\ln \left (10 \ln \left (x \right )-\frac {5 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}+{\mathrm e}^{x}\right )^{4}}}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 18, normalized size = 0.75 \begin {gather*} x + e^{\left (x - e^{\left (\log \left (e^{x} + 10 \, \log \left (x\right )\right )^{4}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 20, normalized size = 0.83 \begin {gather*} x + e^{\left (x - e^{\left (\log \left (e^{x} + 5 \, \log \left (x^{2}\right )\right )^{4}\right )}\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 = 3.74, size = 19, normalized size = 0.79 \begin {gather*} x+{\mathrm {e}}^{-{\mathrm {e}}^{{\ln \left (\ln \left (x^{10}\right )+{\mathrm {e}}^x\right )}^4}}\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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