Optimal. Leaf size=25 \[ e^{e^{2 x}} \left (2+\frac {x}{-3+3 \log \left (\log \left (9+x^2\right )\right )}\right ) \]
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Rubi [F] time = 5.62, 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^{e^{2 x}} \left (-2 x^2+\left (-9-x^2+e^{2 x} \left (108-18 x+12 x^2-2 x^3\right )\right ) \log \left (9+x^2\right )\right )+e^{e^{2 x}} \left (9+x^2+e^{2 x} \left (-216+18 x-24 x^2+2 x^3\right )\right ) \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )+e^{e^{2 x}+2 x} \left (108+12 x^2\right ) \log \left (9+x^2\right ) \log ^2\left (\log \left (9+x^2\right )\right )}{\left (27+3 x^2\right ) \log \left (9+x^2\right )+\left (-54-6 x^2\right ) \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )+\left (27+3 x^2\right ) \log \left (9+x^2\right ) \log ^2\left (\log \left (9+x^2\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^{2 x}} \left (-2 x^2+\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right ) \left (1+2 e^{2 x} (-6+x)+12 e^{2 x} \log \left (\log \left (9+x^2\right )\right )\right )\right )}{3 \left (9+x^2\right ) \log \left (9+x^2\right ) \left (1-\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{e^{2 x}} \left (-2 x^2+\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right ) \left (1+2 e^{2 x} (-6+x)+12 e^{2 x} \log \left (\log \left (9+x^2\right )\right )\right )\right )}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (1-\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {2 e^{e^{2 x}+2 x} \left (-6+x+6 \log \left (\log \left (9+x^2\right )\right )\right )}{-1+\log \left (\log \left (9+x^2\right )\right )}+\frac {e^{e^{2 x}} \left (-2 x^2-9 \log \left (9+x^2\right )-x^2 \log \left (9+x^2\right )+9 \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )+x^2 \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )\right )}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{e^{2 x}} \left (-2 x^2-9 \log \left (9+x^2\right )-x^2 \log \left (9+x^2\right )+9 \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )+x^2 \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )\right )}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} \left (-6+x+6 \log \left (\log \left (9+x^2\right )\right )\right )}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx\\ &=\frac {1}{3} \int \frac {e^{e^{2 x}} \left (-1-\frac {2 x^2}{\left (9+x^2\right ) \log \left (9+x^2\right )}+\log \left (\log \left (9+x^2\right )\right )\right )}{\left (1-\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \left (6 e^{e^{2 x}+2 x}+\frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )}\right ) \, dx\\ &=\frac {1}{3} \int \left (-\frac {2 e^{e^{2 x}} x^2}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}+\frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )}\right ) \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx+4 \int e^{e^{2 x}+2 x} \, dx\\ &=\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{e^{2 x}} x^2}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx+2 \operatorname {Subst}\left (\int e^x \, dx,x,e^{2 x}\right )\\ &=2 e^{e^{2 x}}+\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \left (\frac {e^{e^{2 x}}}{\log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}-\frac {9 e^{e^{2 x}}}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}\right ) \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx\\ &=2 e^{e^{2 x}}+\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{e^{2 x}}}{\log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx+6 \int \frac {e^{e^{2 x}}}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx\\ &=2 e^{e^{2 x}}+\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{e^{2 x}}}{\log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx+6 \int \left (\frac {i e^{e^{2 x}}}{6 (3 i-x) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}+\frac {i e^{e^{2 x}}}{6 (3 i+x) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}\right ) \, dx\\ &=2 e^{e^{2 x}}+i \int \frac {e^{e^{2 x}}}{(3 i-x) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+i \int \frac {e^{e^{2 x}}}{(3 i+x) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{e^{2 x}}}{\log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 26, normalized size = 1.04 \begin {gather*} \frac {1}{3} e^{e^{2 x}} \left (6+\frac {x}{-1+\log \left (\log \left (9+x^2\right )\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 55, normalized size = 2.20 \begin {gather*} \frac {{\left (x - 6\right )} e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} + 6 \, e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} \log \left (\log \left (x^{2} + 9\right )\right )}{3 \, {\left (e^{\left (2 \, x\right )} \log \left (\log \left (x^{2} + 9\right )\right ) - e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 64, normalized size = 2.56 \begin {gather*} \frac {x e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} + 6 \, e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} \log \left (\log \left (x^{2} + 9\right )\right ) - 6 \, e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )}}{3 \, {\left (e^{\left (2 \, x\right )} \log \left (\log \left (x^{2} + 9\right )\right ) - e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 28, normalized size = 1.12
method | result | size |
risch | \(2 \,{\mathrm e}^{{\mathrm e}^{2 x}}+\frac {x \,{\mathrm e}^{{\mathrm e}^{2 x}}}{3 \ln \left (\ln \left (x^{2}+9\right )\right )-3}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 37, normalized size = 1.48 \begin {gather*} \frac {{\left (x - 6\right )} e^{\left (e^{\left (2 \, x\right )}\right )} + 6 \, e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (\log \left (x^{2} + 9\right )\right )}{3 \, {\left (\log \left (\log \left (x^{2} + 9\right )\right ) - 1\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 {{\mathrm {e}}^{2\,x+{\mathrm {e}}^{2\,x}}\,\ln \left (x^2+9\right )\,\left (12\,x^2+108\right )\,{\ln \left (\ln \left (x^2+9\right )\right )}^2+{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\ln \left (x^2+9\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (2\,x^3-24\,x^2+18\,x-216\right )+x^2+9\right )\,\ln \left (\ln \left (x^2+9\right )\right )-{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\left (\ln \left (x^2+9\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (2\,x^3-12\,x^2+18\,x-108\right )+x^2+9\right )+2\,x^2\right )}{\ln \left (x^2+9\right )\,\left (3\,x^2+27\right )\,{\ln \left (\ln \left (x^2+9\right )\right )}^2-\ln \left (x^2+9\right )\,\left (6\,x^2+54\right )\,\ln \left (\ln \left (x^2+9\right )\right )+\ln \left (x^2+9\right )\,\left (3\,x^2+27\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 31, normalized size = 1.24 \begin {gather*} \frac {\left (x + 6 \log {\left (\log {\left (x^{2} + 9 \right )} \right )} - 6\right ) e^{e^{2 x}}}{3 \log {\left (\log {\left (x^{2} + 9 \right )} \right )} - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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