Optimal. Leaf size=27 \[ e^{2 e^{\frac {1}{3} \left (-3+\log \left (x-\frac {3 x \log (x)}{e^4+x}\right )\right )}} \]
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Rubi [F] time = 46.12, 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 (2 \exp \left (\frac {1}{3} \left (-3+\log \left (\frac {e^4 x+x^2-3 x \log (x)}{e^4+x}\right )\right )\right )+\frac {1}{3} \left (-3+\log \left (\frac {e^4 x+x^2-3 x \log (x)}{e^4+x}\right )\right )\right ) \left (-2 e^8+e^4 (6-4 x)+6 x-2 x^2+6 e^4 \log (x)\right )}{-3 e^8 x-6 e^4 x^2-3 x^3+\left (9 e^4 x+9 x^2\right ) \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-1+\frac {2 \sqrt [3]{\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}}}{e}} \left (e^8+(-3+x) x+e^4 (-3+2 x)-3 e^4 \log (x)\right )}{3 \left (e^4+x\right )^2 \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}} \, dx\\ &=\frac {2}{3} \int \frac {e^{-1+\frac {2 \sqrt [3]{\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}}}{e}} \left (e^8+(-3+x) x+e^4 (-3+2 x)-3 e^4 \log (x)\right )}{\left (e^4+x\right )^2 \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}} \, dx\\ &=\frac {\left (2 x^{2/3} \left (e^4+x-3 \log (x)\right )^{2/3}\right ) \int \frac {e^{-1+\frac {2 \sqrt [3]{\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}}}{e}} \left (e^8+(-3+x) x+e^4 (-3+2 x)-3 e^4 \log (x)\right )}{x^{2/3} \left (e^4+x\right )^{4/3} \left (e^4+x-3 \log (x)\right )^{2/3}} \, dx}{3 \left (e^4+x\right )^{2/3} \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}}\\ &=\frac {\left (2 x^{2/3} \left (e^4+x-3 \log (x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\exp \left (-1+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right ) \left (e^8+x^3 \left (-3+x^3\right )+e^4 \left (-3+2 x^3\right )-3 e^4 \log \left (x^3\right )\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (e^4+x\right )^{2/3} \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}}\\ &=\frac {\left (2 x^{2/3} \left (e^4+x-3 \log (x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {\exp \left (7+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}}+\frac {\exp \left (-1+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right ) x^3 \left (-3+x^3\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}}+\frac {\exp \left (3+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right ) \left (-3+2 x^3\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}}-\frac {3 \exp \left (3+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right ) \log \left (x^3\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (e^4+x\right )^{2/3} \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}}\\ &=\frac {\left (2 x^{2/3} \left (e^4+x-3 \log (x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\exp \left (7+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (e^4+x\right )^{2/3} \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}}+\frac {\left (2 x^{2/3} \left (e^4+x-3 \log (x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\exp \left (-1+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right ) x^3 \left (-3+x^3\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (e^4+x\right )^{2/3} \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}}+\frac {\left (2 x^{2/3} \left (e^4+x-3 \log (x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\exp \left (3+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right ) \left (-3+2 x^3\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (e^4+x\right )^{2/3} \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}}-\frac {\left (6 x^{2/3} \left (e^4+x-3 \log (x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\exp \left (3+\frac {2 \sqrt [3]{\frac {x^3 \left (e^4+x^3-3 \log \left (x^3\right )\right )}{e^4+x^3}}}{e}\right ) \log \left (x^3\right )}{\left (e^4+x^3\right )^{4/3} \left (e^4+x^3-3 \log \left (x^3\right )\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (e^4+x\right )^{2/3} \left (\frac {x \left (e^4+x-3 \log (x)\right )}{e^4+x}\right )^{2/3}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F] time = 0.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{2 e^{\frac {1}{3} \left (-3+\log \left (\frac {e^4 x+x^2-3 x \log (x)}{e^4+x}\right )\right )}+\frac {1}{3} \left (-3+\log \left (\frac {e^4 x+x^2-3 x \log (x)}{e^4+x}\right )\right )} \left (-2 e^8+e^4 (6-4 x)+6 x-2 x^2+6 e^4 \log (x)\right )}{-3 e^8 x-6 e^4 x^2-3 x^3+\left (9 e^4 x+9 x^2\right ) \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 308, normalized size = 11.41
method | result | size |
risch | \({\mathrm e}^{\frac {2 x^{\frac {1}{3}} \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )^{\frac {1}{3}} {\mathrm e}^{-1-\frac {i \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right )^{3}}{6}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x +{\mathrm e}^{4}}\right )}{6}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )\right )}{6}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right ) \mathrm {csgn}\left (\frac {i}{x +{\mathrm e}^{4}}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )\right )}{6}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right ) \mathrm {csgn}\left (\frac {i x \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right )^{2}}{6}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right ) \mathrm {csgn}\left (\frac {i x \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right ) \mathrm {csgn}\left (i x \right )}{6}-\frac {i \pi \mathrm {csgn}\left (\frac {i x \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right )^{3}}{6}+\frac {i \pi \mathrm {csgn}\left (\frac {i x \left ({\mathrm e}^{4}+x -3 \ln \relax (x )\right )}{x +{\mathrm e}^{4}}\right )^{2} \mathrm {csgn}\left (i x \right )}{6}}}{\left (x +{\mathrm e}^{4}\right )^{\frac {1}{3}}}}\) | \(308\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 24, normalized size = 0.89 \begin {gather*} e^{\left (\frac {2 \, {\left (x + e^{4} - 3 \, \log \relax (x)\right )}^{\frac {1}{3}} x^{\frac {1}{3}} e^{\left (-1\right )}}{{\left (x + e^{4}\right )}^{\frac {1}{3}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{\frac {\ln \left (\frac {x\,{\mathrm {e}}^4-3\,x\,\ln \relax (x)+x^2}{x+{\mathrm {e}}^4}\right )}{3}-1}}\,{\mathrm {e}}^{\frac {\ln \left (\frac {x\,{\mathrm {e}}^4-3\,x\,\ln \relax (x)+x^2}{x+{\mathrm {e}}^4}\right )}{3}-1}\,\left (2\,{\mathrm {e}}^8-6\,x-6\,{\mathrm {e}}^4\,\ln \relax (x)+2\,x^2+{\mathrm {e}}^4\,\left (4\,x-6\right )\right )}{3\,x\,{\mathrm {e}}^8-\ln \relax (x)\,\left (9\,x^2+9\,{\mathrm {e}}^4\,x\right )+6\,x^2\,{\mathrm {e}}^4+3\,x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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