Optimal. Leaf size=22 \[ e^{x^4-\frac {5}{\log (4 x \log (x))}} \sqrt {x} \]
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Rubi [F]
time = 2.62, 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 (\frac {-10+\left (2 x^4+\log (x)\right ) \log (4 x \log (x))}{2 \log (4 x \log (x))}\right ) \left (10+10 \log (x)+\left (1+8 x^4\right ) \log (x) \log ^2(4 x \log (x))\right )}{2 x \log (x) \log ^2(4 x \log (x))} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {\exp \left (\frac {-10+\left (2 x^4+\log (x)\right ) \log (4 x \log (x))}{2 \log (4 x \log (x))}\right ) \left (10+10 \log (x)+\left (1+8 x^4\right ) \log (x) \log ^2(4 x \log (x))\right )}{x \log (x) \log ^2(4 x \log (x))} \, dx\\ &=\frac {1}{2} \int \left (\frac {\exp \left (\frac {-10+\left (2 x^4+\log (x)\right ) \log (4 x \log (x))}{2 \log (4 x \log (x))}\right ) \left (1+8 x^4\right )}{x}+\frac {10 \exp \left (\frac {-10+\left (2 x^4+\log (x)\right ) \log (4 x \log (x))}{2 \log (4 x \log (x))}\right ) (1+\log (x))}{x \log (x) \log ^2(4 x \log (x))}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {-10+\left (2 x^4+\log (x)\right ) \log (4 x \log (x))}{2 \log (4 x \log (x))}\right ) \left (1+8 x^4\right )}{x} \, dx+5 \int \frac {\exp \left (\frac {-10+\left (2 x^4+\log (x)\right ) \log (4 x \log (x))}{2 \log (4 x \log (x))}\right ) (1+\log (x))}{x \log (x) \log ^2(4 x \log (x))} \, dx\\ &=\frac {1}{2} \int \frac {e^{x^4-\frac {5}{\log (4 x \log (x))}} \left (1+8 x^4\right )}{\sqrt {x}} \, dx+5 \int \frac {e^{x^4-\frac {5}{\log (4 x \log (x))}} (1+\log (x))}{\sqrt {x} \log (x) \log ^2(4 x \log (x))} \, dx\\ &=10 \text {Subst}\left (\int \frac {e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}} \left (1+\log \left (x^2\right )\right )}{\log \left (x^2\right ) \log ^2\left (4 x^2 \log \left (x^2\right )\right )} \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}} \left (1+8 x^8\right ) \, dx,x,\sqrt {x}\right )\\ &=10 \text {Subst}\left (\int \left (\frac {e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}}}{\log ^2\left (4 x^2 \log \left (x^2\right )\right )}+\frac {e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}}}{\log \left (x^2\right ) \log ^2\left (4 x^2 \log \left (x^2\right )\right )}\right ) \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \left (e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}}+8 e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}} x^8\right ) \, dx,x,\sqrt {x}\right )\\ &=8 \text {Subst}\left (\int e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}} x^8 \, dx,x,\sqrt {x}\right )+10 \text {Subst}\left (\int \frac {e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}}}{\log ^2\left (4 x^2 \log \left (x^2\right )\right )} \, dx,x,\sqrt {x}\right )+10 \text {Subst}\left (\int \frac {e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}}}{\log \left (x^2\right ) \log ^2\left (4 x^2 \log \left (x^2\right )\right )} \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int e^{x^8-\frac {5}{\log \left (4 x^2 \log \left (x^2\right )\right )}} \, dx,x,\sqrt {x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.32, size = 22, normalized size = 1.00 \begin {gather*} e^{x^4-\frac {5}{\log (4 x \log (x))}} \sqrt {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 = 7.09, size = 299, normalized size = 13.59
method | result | size |
risch | \({\mathrm e}^{\frac {-2 i \pi \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{3} x^{4}+2 i \pi \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i x \right ) x^{4}+2 i \pi \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right ) x^{4}-2 i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (x \right )\right ) x^{4}-i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{3}+i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right )-i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (x \right )\right )+4 x^{4} \ln \left (x \right )+8 x^{4} \ln \left (2\right )+4 \ln \left (\ln \left (x \right )\right ) x^{4}+2 \ln \left (x \right )^{2}+4 \ln \left (2\right ) \ln \left (x \right )+2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-20}{-2 i \pi \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{3}+2 i \pi \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i x \right )+2 i \pi \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (x \right )\right )-2 i \pi \,\mathrm {csgn}\left (i x \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (x \right )\right )+4 \ln \left (x \right )+8 \ln \left (2\right )+4 \ln \left (\ln \left (x \right )\right )}}\) | \(299\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 28, normalized size = 1.27 \begin {gather*} e^{\left (\frac {{\left (2 \, x^{4} + \log \left (x\right )\right )} \log \left (4 \, x \log \left (x\right )\right ) - 10}{2 \, \log \left (4 \, x \log \left (x\right )\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 29, normalized size = 1.32 \begin {gather*} e^{\frac {\frac {\left (2 x^{4} + \log {\left (x \right )}\right ) \log {\left (4 x \log {\left (x \right )} \right )}}{2} - 5}{\log {\left (4 x \log {\left (x \right )} \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 19, normalized size = 0.86 \begin {gather*} e^{\left (x^{4} - \frac {5}{\log \left (4 \, x \log \left (x\right )\right )} + \frac {1}{2} \, \log \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.84, size = 62, normalized size = 2.82 \begin {gather*} x^{\frac {\ln \left (2\right )}{\ln \left (4\,x\,\ln \left (x\right )\right )}}\,{\mathrm {e}}^{\frac {\ln \left (x\,\ln \left (x\right )\right )\,\ln \left (\sqrt {x}\right )}{\ln \left (4\,x\,\ln \left (x\right )\right )}-\frac {5}{\ln \left (4\,x\,\ln \left (x\right )\right )}}\,{\left (4\,x\,\ln \left (x\right )\right )}^{\frac {x^4}{\ln \left (4\,x\,\ln \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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