Optimal. Leaf size=30 \[ \frac {1}{4} e^{-\frac {1}{2 \left (-\log \left (\frac {3}{2}\right )+e^{2 x} \log (2)\right )}} x^2 \]
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Rubi [F] time = 5.17, 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^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} \left (2 x \log ^2\left (\frac {3}{2}\right )+e^{2 x} \left (x^2-4 x \log \left (\frac {3}{2}\right )\right ) \log (2)+2 e^{4 x} x \log ^2(2)\right )}{4 \log ^2\left (\frac {3}{2}\right )-8 e^{2 x} \log \left (\frac {3}{2}\right ) \log (2)+4 e^{4 x} \log ^2(2)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} \left (2 x \log ^2\left (\frac {3}{2}\right )+e^{2 x} \left (x^2-4 x \log \left (\frac {3}{2}\right )\right ) \log (2)+2 e^{4 x} x \log ^2(2)\right )}{4 \left (\log \left (\frac {3}{2}\right )-e^{2 x} \log (2)\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} \left (2 x \log ^2\left (\frac {3}{2}\right )+e^{2 x} \left (x^2-4 x \log \left (\frac {3}{2}\right )\right ) \log (2)+2 e^{4 x} x \log ^2(2)\right )}{\left (\log \left (\frac {3}{2}\right )-e^{2 x} \log (2)\right )^2} \, dx\\ &=\frac {1}{4} \int \left (2 e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} x+\frac {e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} x^2 \log \left (\frac {3}{2}\right )}{\left (\log \left (\frac {3}{2}\right )-e^{2 x} \log (2)\right )^2}-\frac {e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} x^2}{\log \left (\frac {3}{2}\right )-e^{2 x} \log (2)}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} x^2}{\log \left (\frac {3}{2}\right )-e^{2 x} \log (2)} \, dx\right )+\frac {1}{2} \int e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} x \, dx+\frac {1}{4} \log \left (\frac {3}{2}\right ) \int \frac {e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} x^2}{\left (\log \left (\frac {3}{2}\right )-e^{2 x} \log (2)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F] time = 1.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-\frac {1}{-2 \log \left (\frac {3}{2}\right )+2 e^{2 x} \log (2)}} \left (2 x \log ^2\left (\frac {3}{2}\right )+e^{2 x} \left (x^2-4 x \log \left (\frac {3}{2}\right )\right ) \log (2)+2 e^{4 x} x \log ^2(2)\right )}{4 \log ^2\left (\frac {3}{2}\right )-8 e^{2 x} \log \left (\frac {3}{2}\right ) \log (2)+4 e^{4 x} \log ^2(2)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.07, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, x^{2} e^{\left (-\frac {1}{2 \, {\left (e^{\left (2 \, x\right )} \log \relax (2) - \log \left (\frac {3}{2}\right )\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 114, normalized size = 3.80 \begin {gather*} \frac {1}{4} \, x^{2} e^{\left (2 \, x - \frac {4 \, x e^{\left (2 \, x\right )} \log \relax (3) \log \relax (2) - 4 \, x e^{\left (2 \, x\right )} \log \relax (2)^{2} - 4 \, x \log \relax (3)^{2} + 8 \, x \log \relax (3) \log \relax (2) - 4 \, x \log \relax (2)^{2} + e^{\left (2 \, x\right )} \log \relax (2)}{2 \, {\left (e^{\left (2 \, x\right )} \log \relax (3) \log \relax (2) - e^{\left (2 \, x\right )} \log \relax (2)^{2} - \log \relax (3)^{2} + 2 \, \log \relax (3) \log \relax (2) - \log \relax (2)^{2}\right )}} + \frac {1}{2 \, {\left (\log \relax (3) - \log \relax (2)\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 25, normalized size = 0.83
method | result | size |
risch | \(\frac {x^{2} {\mathrm e}^{-\frac {1}{2 \left (\ln \relax (2) {\mathrm e}^{2 x}+\ln \relax (2)-\ln \relax (3)\right )}}}{4}\) | \(25\) |
norman | \(\frac {\left (\left (\frac {\ln \relax (2)}{4}-\frac {\ln \relax (3)}{4}\right ) x^{2}+\frac {x^{2} \ln \relax (2) {\mathrm e}^{2 x}}{4}\right ) {\mathrm e}^{-\frac {1}{{\mathrm e}^{2 x} \ln \relax (4)+\ln \left (\frac {4}{9}\right )}}}{\ln \relax (2) {\mathrm e}^{2 x}-\ln \left (\frac {3}{2}\right )}\) | \(59\) |
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*} \frac {1}{4} \, \int \frac {{\left (2 \, x e^{\left (4 \, x\right )} \log \relax (2)^{2} + {\left (x^{2} - 4 \, x \log \left (\frac {3}{2}\right )\right )} e^{\left (2 \, x\right )} \log \relax (2) + 2 \, x \log \left (\frac {3}{2}\right )^{2}\right )} e^{\left (-\frac {1}{2 \, {\left (e^{\left (2 \, x\right )} \log \relax (2) - \log \left (\frac {3}{2}\right )\right )}}\right )}}{e^{\left (4 \, x\right )} \log \relax (2)^{2} - 2 \, e^{\left (2 \, x\right )} \log \relax (2) \log \left (\frac {3}{2}\right ) + \log \left (\frac {3}{2}\right )^{2}}\,{d x} \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 = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {1}{2\,\ln \left (\frac {3}{2}\right )-2\,{\mathrm {e}}^{2\,x}\,\ln \relax (2)}}\,\left (2\,x\,{\ln \left (\frac {3}{2}\right )}^2+2\,x\,{\mathrm {e}}^{4\,x}\,{\ln \relax (2)}^2-{\mathrm {e}}^{2\,x}\,\ln \relax (2)\,\left (4\,x\,\ln \left (\frac {3}{2}\right )-x^2\right )\right )}{4\,{\mathrm {e}}^{4\,x}\,{\ln \relax (2)}^2+4\,{\ln \left (\frac {3}{2}\right )}^2-8\,{\mathrm {e}}^{2\,x}\,\ln \relax (2)\,\ln \left (\frac {3}{2}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.87, size = 24, normalized size = 0.80 \begin {gather*} \frac {x^{2} e^{- \frac {1}{2 e^{2 x} \log {\relax (2 )} - 2 \log {\left (\frac {3}{2} \right )}}}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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