Optimal. Leaf size=25 \[ x+e^{\frac {e^{-4 x^2} x}{\log (x)}} (4-5 (7+x)) \]
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Rubi [F]
time = 3.32, 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^{-4 x^2} \left (e^{4 x^2} \log ^2(x)+e^{\frac {e^{-4 x^2} x}{\log (x)}} \left (31+5 x+\left (-31-5 x+248 x^2+40 x^3\right ) \log (x)-5 e^{4 x^2} \log ^2(x)\right )\right )}{\log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-5 e^{\frac {e^{-4 x^2} x}{\log (x)}}+\frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} (31+5 x) \left (1-\log (x)+8 x^2 \log (x)\right )}{\log ^2(x)}\right ) \, dx\\ &=x-5 \int e^{\frac {e^{-4 x^2} x}{\log (x)}} \, dx+\int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} (31+5 x) \left (1-\log (x)+8 x^2 \log (x)\right )}{\log ^2(x)} \, dx\\ &=x-5 \int e^{\frac {e^{-4 x^2} x}{\log (x)}} \, dx+\int \left (\frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} (31+5 x)}{\log ^2(x)}+\frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} \left (-31-5 x+248 x^2+40 x^3\right )}{\log (x)}\right ) \, dx\\ &=x-5 \int e^{\frac {e^{-4 x^2} x}{\log (x)}} \, dx+\int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} (31+5 x)}{\log ^2(x)} \, dx+\int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} \left (-31-5 x+248 x^2+40 x^3\right )}{\log (x)} \, dx\\ &=x-5 \int e^{\frac {e^{-4 x^2} x}{\log (x)}} \, dx+\int \left (\frac {31 e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}}}{\log ^2(x)}+\frac {5 e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} x}{\log ^2(x)}\right ) \, dx+\int \left (-\frac {31 e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}}}{\log (x)}-\frac {5 e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} x}{\log (x)}+\frac {248 e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} x^2}{\log (x)}+\frac {40 e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} x^3}{\log (x)}\right ) \, dx\\ &=x-5 \int e^{\frac {e^{-4 x^2} x}{\log (x)}} \, dx+5 \int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} x}{\log ^2(x)} \, dx-5 \int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} x}{\log (x)} \, dx+31 \int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}}}{\log ^2(x)} \, dx-31 \int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}}}{\log (x)} \, dx+40 \int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} x^3}{\log (x)} \, dx+248 \int \frac {e^{-4 x^2+\frac {e^{-4 x^2} x}{\log (x)}} x^2}{\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.69, size = 23, normalized size = 0.92 \begin {gather*} e^{\frac {e^{-4 x^2} x}{\log (x)}} (-31-5 x)+x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 22, normalized size = 0.88
method | result | size |
risch | \(x +{\mathrm e}^{\frac {x \,{\mathrm e}^{-4 x^{2}}}{\ln \left (x \right )}} \left (-31-5 x \right )\) | \(22\) |
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 = 0.39, size = 22, normalized size = 0.88 \begin {gather*} -{\left (5 \, x + 31\right )} e^{\left (\frac {x e^{\left (-4 \, x^{2}\right )}}{\log \left (x\right )}\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{-4\,x^2}\,\left ({\mathrm {e}}^{4\,x^2}\,{\ln \left (x\right )}^2+{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{-4\,x^2}}{\ln \left (x\right )}}\,\left (-5\,{\mathrm {e}}^{4\,x^2}\,{\ln \left (x\right )}^2+\left (40\,x^3+248\,x^2-5\,x-31\right )\,\ln \left (x\right )+5\,x+31\right )\right )}{{\ln \left (x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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