Optimal. Leaf size=28 \[ \frac {1}{2} \left (-1-e^{e^{\frac {2 e^{-x^4}}{\log (5)}}}-x\right ) \]
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Rubi [A] time = 0.53, antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 5, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {12, 6742, 6715, 2282, 2194} \begin {gather*} -\frac {1}{2} e^{e^{\frac {2 e^{-x^4}}{\log (5)}}}-\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2194
Rule 2282
Rule 6715
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int e^{-x^4} \left (8 e^{e^{\frac {2 e^{-x^4}}{\log (5)}}+\frac {2 e^{-x^4}}{\log (5)}} x^3-e^{x^4} \log (5)\right ) \, dx}{2 \log (5)}\\ &=\frac {\int \left (8 e^{e^{\frac {2 e^{-x^4}}{\log (5)}}-x^4+\frac {2 e^{-x^4}}{\log (5)}} x^3-\log (5)\right ) \, dx}{2 \log (5)}\\ &=-\frac {x}{2}+\frac {4 \int e^{e^{\frac {2 e^{-x^4}}{\log (5)}}-x^4+\frac {2 e^{-x^4}}{\log (5)}} x^3 \, dx}{\log (5)}\\ &=-\frac {x}{2}+\frac {\operatorname {Subst}\left (\int e^{e^{\frac {2 e^{-x}}{\log (5)}}-x+\frac {2 e^{-x}}{\log (5)}} \, dx,x,x^4\right )}{\log (5)}\\ &=-\frac {x}{2}-\frac {\operatorname {Subst}\left (\int e^{e^{\frac {2 x}{\log (5)}}+\frac {2 x}{\log (5)}} \, dx,x,e^{-x^4}\right )}{\log (5)}\\ &=-\frac {x}{2}-\frac {1}{2} \operatorname {Subst}\left (\int e^x \, dx,x,e^{\frac {2 e^{-x^4}}{\log (5)}}\right )\\ &=-\frac {1}{2} e^{e^{\frac {2 e^{-x^4}}{\log (5)}}}-\frac {x}{2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 32, normalized size = 1.14 \begin {gather*} \frac {-e^{e^{\frac {2 e^{-x^4}}{\log (5)}}} \log (5)-x \log (5)}{\log (25)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 65, normalized size = 2.32 \begin {gather*} -\frac {1}{2} \, {\left (x e^{\left (\frac {2 \, e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )} + e^{\left (\frac {{\left (e^{\left (x^{4} + \frac {2 \, e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )} \log \relax (5) + 2\right )} e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )}\right )} e^{\left (-\frac {2 \, e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )} \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*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 0.75
method | result | size |
risch | \(-\frac {x}{2}-\frac {{\mathrm e}^{{\mathrm e}^{\frac {2 \,{\mathrm e}^{-x^{4}}}{\ln \relax (5)}}}}{2}\) | \(21\) |
default | \(\frac {-{\mathrm e}^{{\mathrm e}^{\frac {2 \,{\mathrm e}^{-x^{4}}}{\ln \relax (5)}}} \ln \relax (5)-x \ln \relax (5)}{2 \ln \relax (5)}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 28, normalized size = 1.00 \begin {gather*} -\frac {x \log \relax (5) + e^{\left (e^{\left (\frac {2 \, e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )}\right )} \log \relax (5)}{2 \, \log \relax (5)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.98, size = 20, normalized size = 0.71 \begin {gather*} -\frac {x}{2}-\frac {{\mathrm {e}}^{{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{-x^4}}{\ln \relax (5)}}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 19, normalized size = 0.68 \begin {gather*} - \frac {x}{2} - \frac {e^{e^{\frac {2 e^{- x^{4}}}{\log {\relax (5 )}}}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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