Optimal. Leaf size=26 \[ 3 e^{-x} \left (x+x^2\right ) \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right ) \]
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Rubi [A] time = 1.08, antiderivative size = 47, normalized size of antiderivative = 1.81, number of steps used = 18, number of rules used = 6, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {6741, 6742, 2194, 2176, 2196, 2555} \begin {gather*} 3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (\log \left (x^2\right )+3\right )}{x}\right )+3 e^{-x} x \log (3) \log \left (\frac {3 \left (\log \left (x^2\right )+3\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2196
Rule 2555
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left ((-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )\right )}{3+\log \left (x^2\right )} \, dx\\ &=\int \left (-\frac {3 e^{-x} (1+x) \log (3) \left (1+\log \left (x^2\right )\right )}{3+\log \left (x^2\right )}-3 e^{-x} \left (-1-x+x^2\right ) \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )\right ) \, dx\\ &=-\left ((3 \log (3)) \int \frac {e^{-x} (1+x) \left (1+\log \left (x^2\right )\right )}{3+\log \left (x^2\right )} \, dx\right )-(3 \log (3)) \int e^{-x} \left (-1-x+x^2\right ) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right ) \, dx\\ &=3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+(3 \log (3)) \int \frac {e^{-x} (1+x) \left (1+\log \left (x^2\right )\right )}{3+\log \left (x^2\right )} \, dx-(3 \log (3)) \int \left (e^{-x}+e^{-x} x-\frac {2 e^{-x} (1+x)}{3+\log \left (x^2\right )}\right ) \, dx\\ &=3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )-(3 \log (3)) \int e^{-x} \, dx-(3 \log (3)) \int e^{-x} x \, dx+(3 \log (3)) \int \left (e^{-x}+e^{-x} x-\frac {2 e^{-x} (1+x)}{3+\log \left (x^2\right )}\right ) \, dx+(6 \log (3)) \int \frac {e^{-x} (1+x)}{3+\log \left (x^2\right )} \, dx\\ &=3 e^{-x} \log (3)+3 e^{-x} x \log (3)+3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+(3 \log (3)) \int e^{-x} x \, dx-(6 \log (3)) \int \frac {e^{-x} (1+x)}{3+\log \left (x^2\right )} \, dx+(6 \log (3)) \int \left (\frac {e^{-x}}{3+\log \left (x^2\right )}+\frac {e^{-x} x}{3+\log \left (x^2\right )}\right ) \, dx\\ &=3 e^{-x} \log (3)+3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+(3 \log (3)) \int e^{-x} \, dx+(6 \log (3)) \int \frac {e^{-x}}{3+\log \left (x^2\right )} \, dx+(6 \log (3)) \int \frac {e^{-x} x}{3+\log \left (x^2\right )} \, dx-(6 \log (3)) \int \left (\frac {e^{-x}}{3+\log \left (x^2\right )}+\frac {e^{-x} x}{3+\log \left (x^2\right )}\right ) \, dx\\ &=3 e^{-x} x \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )+3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.55, size = 25, normalized size = 0.96 \begin {gather*} 3 e^{-x} x (1+x) \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 25, normalized size = 0.96 \begin {gather*} 3 \, {\left (x^{2} + x\right )} e^{\left (-x\right )} \log \relax (3) \log \left (\frac {3 \, {\left (\log \left (x^{2}\right ) + 3\right )}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 83, normalized size = 3.19 \begin {gather*} 3 \, x^{2} e^{\left (-x\right )} \log \relax (3)^{2} - 3 \, x^{2} e^{\left (-x\right )} \log \relax (3) \log \relax (x) + 3 \, x^{2} e^{\left (-x\right )} \log \relax (3) \log \left (\log \left (x^{2}\right ) + 3\right ) + 3 \, x e^{\left (-x\right )} \log \relax (3)^{2} - 3 \, x e^{\left (-x\right )} \log \relax (3) \log \relax (x) + 3 \, x e^{\left (-x\right )} \log \relax (3) \log \left (\log \left (x^{2}\right ) + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 1616, normalized size = 62.15
method | result | size |
risch | \(3 \ln \relax (3) x \left (x +1\right ) {\mathrm e}^{-x} \ln \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )+\frac {3 \ln \relax (3) x \left (-i \pi \,\mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )+i x \pi \,\mathrm {csgn}\left (i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{2}+i x \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{2}-i x \pi \,\mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )-i x \pi \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{3}-i x \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )-i x \pi +i x \pi \,\mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right ) \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{3}-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )+i \pi \,\mathrm {csgn}\left (i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{3}+i x \pi \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{x}\right )^{2}-i x \pi \mathrm {csgn}\left (\frac {i \left (6 i+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}\right )}{x}\right )^{3}-i \pi +2 x \ln \left (\frac {3}{2}\right )-2 x \ln \relax (x )+2 \ln \left (\frac {3}{2}\right )-2 \ln \relax (x )\right ) {\mathrm e}^{-x}}{2}\) | \(1616\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 61, normalized size = 2.35 \begin {gather*} 3 \, {\left (x^{2} \log \relax (3) + x \log \relax (3)\right )} e^{\left (-x\right )} \log \left (2 \, \log \relax (x) + 3\right ) + 3 \, {\left (x^{2} \log \relax (3)^{2} + x \log \relax (3)^{2} - {\left (x^{2} \log \relax (3) + x \log \relax (3)\right )} \log \relax (x)\right )} e^{\left (-x\right )} \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 {\ln \relax (3)\,\left (3\,x+3\right )-\ln \left (\frac {3\,\ln \left (x^2\right )+9}{x}\right )\,\left (\ln \relax (3)\,\left (-9\,x^2+9\,x+9\right )+\ln \left (x^2\right )\,\ln \relax (3)\,\left (-3\,x^2+3\,x+3\right )\right )+\ln \left (x^2\right )\,\ln \relax (3)\,\left (3\,x+3\right )}{3\,{\mathrm {e}}^x+\ln \left (x^2\right )\,{\mathrm {e}}^x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.97, size = 41, normalized size = 1.58 \begin {gather*} \left (3 x^{2} \log {\relax (3 )} \log {\left (\frac {3 \log {\left (x^{2} \right )} + 9}{x} \right )} + 3 x \log {\relax (3 )} \log {\left (\frac {3 \log {\left (x^{2} \right )} + 9}{x} \right )}\right ) e^{- x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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