Optimal. Leaf size=24 \[ x-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)} \]
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Rubi [A] time = 0.09, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} -e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1593
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)\right ) \, dx}{e^2 \log (4)}\\ &=x-\frac {x^2}{\log (4)}+\frac {\int e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \, dx}{e^2}\\ &=x-\frac {x^2}{\log (4)}+\frac {\int e^{\frac {32 x}{e^2}} \left (-2 e^2-32 x\right ) x \, dx}{e^2}\\ &=x-\frac {x^2}{\log (4)}+\frac {\int \left (-2 e^{2+\frac {32 x}{e^2}} x-32 e^{\frac {32 x}{e^2}} x^2\right ) \, dx}{e^2}\\ &=x-\frac {x^2}{\log (4)}-\frac {2 \int e^{2+\frac {32 x}{e^2}} x \, dx}{e^2}-\frac {32 \int e^{\frac {32 x}{e^2}} x^2 \, dx}{e^2}\\ &=x-\frac {1}{16} e^{2+\frac {32 x}{e^2}} x-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)}+\frac {1}{16} \int e^{2+\frac {32 x}{e^2}} \, dx+2 \int e^{\frac {32 x}{e^2}} x \, dx\\ &=\frac {1}{512} e^{4+\frac {32 x}{e^2}}+x-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)}-\frac {1}{16} e^2 \int e^{\frac {32 x}{e^2}} \, dx\\ &=x-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 22, normalized size = 0.92 \begin {gather*} x \left (1-e^{\frac {32 x}{e^2}} x-\frac {x}{\log (4)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 28, normalized size = 1.17 \begin {gather*} -\frac {2 \, x^{2} e^{\left (32 \, x e^{\left (-2\right )}\right )} \log \relax (2) + x^{2} - 2 \, x \log \relax (2)}{2 \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 71, normalized size = 2.96 \begin {gather*} -\frac {{\left (256 \, x^{2} e^{2} - 512 \, x e^{2} \log \relax (2) + {\left ({\left (32 \, x e^{2} - e^{4}\right )} e^{\left (2 \, {\left (16 \, x + e^{2}\right )} e^{\left (-2\right )}\right )} + {\left (512 \, x^{2} e^{2} - 32 \, x e^{4} + e^{6}\right )} e^{\left (32 \, x e^{\left (-2\right )}\right )}\right )} \log \relax (2)\right )} e^{\left (-2\right )}}{512 \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 23, normalized size = 0.96
method | result | size |
risch | \(x -\frac {x^{2}}{2 \ln \relax (2)}-{\mathrm e}^{32 x \,{\mathrm e}^{-2}} x^{2}\) | \(23\) |
derivativedivides | \(\frac {-128 x^{2}+256 x \ln \relax (2)-256 \ln \relax (2) {\mathrm e}^{32 x \,{\mathrm e}^{-2}} x^{2}}{256 \ln \relax (2)}\) | \(35\) |
norman | \(\left (x \,{\mathrm e}-x^{2} {\mathrm e} \,{\mathrm e}^{32 x \,{\mathrm e}^{-2}}-\frac {{\mathrm e} x^{2}}{2 \ln \relax (2)}\right ) {\mathrm e}^{-1}\) | \(39\) |
default | \(\frac {{\mathrm e}^{-2} \left (x \,{\mathrm e}^{2} \ln \relax (2)-{\mathrm e}^{2} \ln \relax (2) {\mathrm e}^{32 x \,{\mathrm e}^{-2}} x^{2}-\frac {x^{2} {\mathrm e}^{2}}{2}\right )}{\ln \relax (2)}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 37, normalized size = 1.54 \begin {gather*} -\frac {{\left (2 \, x^{2} e^{\left (32 \, x e^{\left (-2\right )} + 2\right )} \log \relax (2) + x^{2} e^{2} - 2 \, x e^{2} \log \relax (2)\right )} e^{\left (-2\right )}}{2 \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 22, normalized size = 0.92 \begin {gather*} x-\frac {x^2}{2\,\ln \relax (2)}-x^2\,{\mathrm {e}}^{32\,x\,{\mathrm {e}}^{-2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.83 \begin {gather*} - x^{2} e^{\frac {32 x}{e^{2}}} - \frac {x^{2}}{2 \log {\relax (2 )}} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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