Optimal. Leaf size=26 \[ 3-x+x^2-\left (2+\log \left (3-e^{e^x}+\log (3)\right )\right )^2 \]
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Rubi [A] time = 0.57, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 5, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6741, 6742, 2282, 2390, 2301} \begin {gather*} x^2-x-\left (\log \left (-e^{e^x}+3+\log (3)\right )+2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 2282
Rule 2301
Rule 2390
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3-6 x+e^{e^x} \left (-1-4 e^x+2 x\right )+(1-2 x) \log (3)-2 e^{e^x+x} \log \left (3-e^{e^x}+\log (3)\right )}{e^{e^x}-3 \left (1+\frac {\log (3)}{3}\right )} \, dx\\ &=\int \left (-1+2 x+\frac {2 e^{e^x+x} \left (-2-\log \left (3-e^{e^x}+\log (3)\right )\right )}{e^{e^x}-3 \left (1+\frac {\log (3)}{3}\right )}\right ) \, dx\\ &=-x+x^2+2 \int \frac {e^{e^x+x} \left (-2-\log \left (3-e^{e^x}+\log (3)\right )\right )}{e^{e^x}-3 \left (1+\frac {\log (3)}{3}\right )} \, dx\\ &=-x+x^2+2 \operatorname {Subst}\left (\int \frac {e^x \left (-2-\log \left (3-e^x+\log (3)\right )\right )}{e^x-3 \left (1+\frac {\log (3)}{3}\right )} \, dx,x,e^x\right )\\ &=-x+x^2+2 \operatorname {Subst}\left (\int \frac {2+\log (3-x+\log (3))}{3-x+\log (3)} \, dx,x,e^{e^x}\right )\\ &=-x+x^2-2 \operatorname {Subst}\left (\int \frac {2+\log (x)}{x} \, dx,x,3-e^{e^x}+\log (3)\right )\\ &=-x+x^2-\left (2+\log \left (3-e^{e^x}+\log (3)\right )\right )^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.28, size = 38, normalized size = 1.46 \begin {gather*} -4-x+x^2-4 \log \left (3-e^{e^x}+\log (3)\right )-\log ^2\left (3-e^{e^x}+\log (3)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 55, normalized size = 2.12 \begin {gather*} x^{2} - \log \left ({\left ({\left (\log \relax (3) + 3\right )} e^{x} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )^{2} - x - 4 \, \log \left ({\left ({\left (\log \relax (3) + 3\right )} e^{x} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 33, normalized size = 1.27 \begin {gather*} x^{2} - \log \left (-e^{\left (e^{x}\right )} + \log \relax (3) + 3\right )^{2} - x - 4 \, \log \left (e^{\left (e^{x}\right )} - \log \relax (3) - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 34, normalized size = 1.31
method | result | size |
default | \(x^{2}-x -\ln \left (-{\mathrm e}^{{\mathrm e}^{x}}+3+\ln \relax (3)\right )^{2}-4 \ln \left (-{\mathrm e}^{{\mathrm e}^{x}}+3+\ln \relax (3)\right )\) | \(34\) |
norman | \(x^{2}-x -\ln \left (-{\mathrm e}^{{\mathrm e}^{x}}+3+\ln \relax (3)\right )^{2}-4 \ln \left (-{\mathrm e}^{{\mathrm e}^{x}}+3+\ln \relax (3)\right )\) | \(34\) |
risch | \(-\ln \left (-{\mathrm e}^{{\mathrm e}^{x}}+3+\ln \relax (3)\right )^{2}+x^{2}-x -4 \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-\ln \relax (3)-3\right )\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 33, normalized size = 1.27 \begin {gather*} x^{2} - \log \left (-e^{\left (e^{x}\right )} + \log \relax (3) + 3\right )^{2} - x - 4 \, \log \left (-e^{\left (e^{x}\right )} + \log \relax (3) + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.93, size = 43, normalized size = 1.65 \begin {gather*} x^2-{\ln \left (\ln \relax (3)-{\mathrm {e}}^{{\mathrm {e}}^x}+3\right )}^2-x-\frac {\ln \left ({\mathrm {e}}^{{\mathrm {e}}^x}-\ln \relax (3)-3\right )\,\left (\ln \left (81\right )+12\right )}{\ln \relax (3)+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 31, normalized size = 1.19 \begin {gather*} x^{2} - x - \log {\left (- e^{e^{x}} + \log {\relax (3 )} + 3 \right )}^{2} - 4 \log {\left (e^{e^{x}} - 3 - \log {\relax (3 )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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